Did Bell's Theory Contain a Mathematical Flaw?

[JesseM]

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

OK, so we're rewriting 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) with the assumption that the lambdas only take a discrete range of values. Since subscript is normally used to denote different possible values, let's replace the notation \lambda_1 and \lambda_2 with \lambda_{A,i} and \lambda_{B,j}, where the A subscript refers to local variables associated with the particle whose measurement result is denoted A, and likewise for B. Then if i in \lambda_{A,i} can take any integer value from 1 to N, and j in \lambda_{B,j} can take any integer value from 1 to M, the sum would be:

P(A, B|a, b) = \sum_{i=1}^N \sum_{j=1}^M P_3(\lambda_{A,i}, \lambda_{B,j})*P_1(A|a, \lambda_{A,i})*P_2(B|b, \lambda_{B,j})

 

[Gordon Watson]

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/0912.4098I have more too, but this would probably be a good start for background. I hope this helps.

DrC, continuing --

Thank you for the references.

My approach differs from all Bell-papers known to me.

The new perspective I bring is a critical analysis of Bell's mathematics.

That was a good suggestion by you, thanks.

Do you have any references in that area?

Also, could you provide a reference (for use here at PF - so available to all) to a Mermin-style experiment (gedanken would be best) with two identically correlated photons and the Red and Green lights emitted from just two polarizer-analyser boxes.

Complications can follow later.

I find it best to use that example. The photons fast, the correlations neat.Also, being so opposed to my approach, I think that makes you well qualified to help me with an IR proposal. I believe you must carry some weight ... at PF too.

Regards,

JenniT

 

[Gordon Watson]

OK, so we're rewriting 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) with the assumption that the lambdas only take a discrete range of values. Since subscript is normally used to denote different possible values, let's replace the notation \lambda_1 and \lambda_2 with \lambda_{A,i} and \lambda_{B,j}, where the A subscript refers to local variables associated with the particle whose measurement result is denoted A, and likewise for B. Then if i in \lambda_{A,i} can take any integer value from 1 to N, and j in \lambda_{B,j} can take any integer value from 1 to M, the sum would be:

P(A, B|a, b) = \sum_{i=1}^N \sum_{j=1}^M P_3(\lambda_{A,i}, \lambda_{B,j})*P_1(A|a, \lambda_{A,i})*P_2(B|b, \lambda_{B,j})

JesseM,

Great -- thank you -- you have anticipated my next move -- give me a little time to rewrite in TeX a small refinement -- which I am confident you will accept.

JenniT

 

[Gordon Watson]

OK, so we're rewriting 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) with the assumption that the lambdas only take a discrete range of values. Since subscript is normally used to denote different possible values, let's replace the notation \lambda_1 and \lambda_2 with \lambda_{A,i} and \lambda_{B,j}, where the A subscript refers to local variables associated with the particle whose measurement result is denoted A, and likewise for B. Then if i in \lambda_{A,i} can take any integer value from 1 to N, and j in \lambda_{B,j} can take any integer value from 1 to M, the sum would be:

P(A, B|a, b) = \sum_{i=1}^N \sum_{j=1}^M P_3(\lambda_{A,i}, \lambda_{B,j})*P_1(A|a, \lambda_{A,i})*P_2(B|b, \lambda_{B,j})

JesseM,

Subject to all the preceding formalities, are you OK with these?

It would be good if you would explicate my terms in terms of your QM understanding -- this would be helpful to all the beginners here, I am sure -- noting any area that departs from QM or from Bell.

Condition H simply defines the experiment under consideration: Mermin's boxes, identically correlated photons, primes (') denoting elements of reality associated with Bob.



(A) Local realism defined

P(G, G'|H, a, b', \lambda_{i}, \lambda'_{i}) = P_1(G|H, a, \lambda_{i})*P_2(G'|H, b', \lambda'_{i}).

(B) Expectation defined

P(G, G'|H, a, b') = \sum_{i=1}^N P_3(\lambda_{i}, \lambda'_{i})*P_1(G|H, a, \lambda_{i})*P_2(G'|H, b', \lambda'_{i}).


As my accounting boyfriend says all the time: E & OE.

Love,

JenniT

 

[JesseM]

JesseM,

Subject to all the preceding formalities, are you OK with these?

It would be good if you would explicate my terms in terms of your QM understanding -- this would be helpful to all the beginners here, I am sure -- noting any area that departs from QM or from Bell.

Condition H simply defines the experiment under consideration: Mermin's boxes, identically correlated photons, primes (') denoting elements of reality associated with Bob.



(A) Local realism defined

P(G, G'|H, a, b', \lambda_{i}, \lambda'_{i}) = P_1(G|H, a, \lambda_{i})*P_2(G'|H, b', \lambda'_{i}).

(B) Expectation defined

P(G, G'|H, a, b') = \sum_{i=1}^N P_3(\lambda_{i}, \lambda'_{i})*P_1(G|H, a, \lambda_{i})*P_2(G'|H, b', \lambda'_{i}).

Most of this looks good to me, my only question is about why you made it a sum only over identical values of lambda, rather than a double sum over all possible combinations of values. I would guess this is because you want to include the condition that identical detector settings always imply identical results G and G'? This requires that lambda contain no additional information beyond a set of results for every possible measurement setting--if there were two different values of lambda, say i=12 and i=17, that both implied exactly the same set of results for every possible measurement setting, then it could be that the source would occasionally send out a pair with these different values and it wouldn't conflict with the observation that same settings always give same results. But as long as lambda is defined in a minimal way so that different values of lambda always imply different measurement results for some possible pair of detector settings, then it is safe to assume that the source must always send out pairs that both have the same value of lambda, so a single sum like the one you wrote is fine.

 

[Gordon Watson]

Most of this looks good to me, my only question is about why you made it a sum only over identical values of lambda, rather than a double sum over all possible combinations of values. I would guess this is because you want to include the condition that identical detector settings always imply identical results G and G'? This requires that lambda contain no additional information beyond a set of results for every possible measurement setting--if there were two different values of lambda, say i=12 and i=17, that both implied exactly the same set of results for every possible measurement setting, then it could be that the source would occasionally send out a pair with these different values and it wouldn't conflict with the observation that same settings always give same results. But as long as lambda is defined in a minimal way so that different values of lambda always imply different measurement results for some possible pair of detector settings, then it is safe to assume that the source must always send out pairs that both have the same value of lambda, so a single sum like the one you wrote is fine.

OK - thank you -- you just beat me to lodging this refinement -- an old notation of mine that sits better with the use of subscripts yet to come. I found it hard to continue clearly with identical but unrelated subscripts:

<br /> P&#039;&#039;(G, G&#039;|H, a, b&#039;) = \sum_{i=1}^N P_3(\lambda_{i}, \lambda&#039;_{i})*P(G|H, a, \lambda_{i})*P&#039;(G&#039;|H, b&#039;, \lambda&#039;_{i}).<br />

Trust that's OK.

We next come to defining EPR elements of physical reality.

I will try to get it done during this busy weekend.

Your detailed replies are much appreciated.

JenniT

 

[JesseM]

OK - thank you -- you just beat me to lodging this refinement -- an old notation of mine that sits better with the use of subscripts yet to come. I found it hard to continue clearly with identical but unrelated subscripts:

<br /> P&#039;&#039;(G, G&#039;|H, a, b&#039;) = \sum_{i=1}^N P_3(\lambda_{i}, \lambda&#039;_{i})*P(G|H, a, \lambda_{i})*P&#039;(G&#039;|H, b&#039;, \lambda&#039;_{i}).<br />

Trust that's OK.

Yes, with the conditions I mentioned it's fine.

We next come to defining EPR elements of physical reality.

I will try to get it done during this busy weekend.

Your detailed replies are much appreciated.

JenniT

Take your time, it turns out I'll be taking a trip tomorrow and won't be back until Monday anyway.

 

[Salman2]

We next come to defining EPR elements of physical reality.

fyi, David Bohm (1951, Quantum Theory) discusses the Einstein, Rosen, Podolsky view. Here is summary of ERP argument as understood by Bohm:

1. Every element of physical reality must have a counterpart.

2. If, before measurement the value of a physical quantity can be predicted, then there exists an element of physical reality corresponding to this physical quantity.

These two criteria rest on two assumptions:

3. The universe can be correctly analyzed in terms of distinct and separate "elements of reality"

4. Every element of physical reality must have a counterpart of a precisely defined mathematical quantity.

==

Does this summary by Bohm match your understanding of ERP ?

 

[billschnieder]

fyi, David Bohm (1951, Quantum Theory) discusses the Einstein, Rosen, Podolsky view. Here is summary of ERP argument as understood by Bohm:

1. Every element of physical reality must have a counterpart.

2. If, before measurement the value of a physical quantity can be predicted, then there exists an element of physical reality corresponding to this physical quantity.

These two criteria rest on two assumptions:

3. The universe can be correctly analyzed in terms of distinct and separate "elements of reality"

4. Every element of physical reality must have a counterpart of a precisely defined mathematical quantity.

==

Does this summary by Bohm match your understanding of ERP ?

This is not correct. According to EPR, (1) is a condition to be applied to a complete theory, without any implications for the meaning of "elements of reality". You may have confused the fact that the main focus of EPR was to argue that QM is not complete rather than to define in an exhaustive manner what "reality" means. So the definition they gave which is your (2) above, is an operational definition in the context of trying to prove that QM is not complete. Therefore your claims (1), (3), (4) are not accurate representations of what EPR defined as "reality", only (2) is correct.

 

[Gordon Watson]

fyi, David Bohm (1951, Quantum Theory) discusses the Einstein, Rosen, Podolsky view. Here is summary of ERP argument as understood by Bohm:

1. Every element of physical reality must have a counterpart.

2. If, before measurement the value of a physical quantity can be predicted, then there exists an element of physical reality corresponding to this physical quantity.

These two criteria rest on two assumptions:

3. The universe can be correctly analyzed in terms of distinct and separate "elements of reality"

4. Every element of physical reality must have a counterpart of a precisely defined mathematical quantity.

==

Does this summary by Bohm match your understanding of ERP ?

Dear Salman2,

This is a short quick post to thank you very much for posting these important details.

To put your mind at ease, my short answer is YES, certainly ... BUT ...

1. Thank you for using Bohm's funny ERP!

2. Please see Bill's very important post above.

From my first reading of what Bill writes there, he puts my position exactly and very nicely.

I may need a BUT there too, when I've studied it, but please be clear about the EPR paper.

It is a beauty!

My elements of physical reality are, in my terms, binding spin trajectories associated with the perfect rotational symmetries attaching to fundamental particles.

But I am not a physicist so, if you are, please be very critical of my terminology so that I can improve it.

EDIT: I hope this is not too silly by me but I see some discussion by you on spin: I see fundamental particles as magic gyroscopes subject to binding spin trajectories when disturbed -- elements of physical reality -- the consequence of perfect rotational symmetries. Also this: Their polarizers burn off any extrinsic spin (spin = short term for angular momentum) and re-orient the intrinsic spin. Clarity here would help physicists better understand my theory -- or me correct its terms. Maybe needs separate thread -- for words and diagrams -- this one for mathematics? Thank you again.

Thank you again,

JenniT

 

[Gordon Watson]

This is not correct. According to EPR, (1) is a condition to be applied to a complete theory, without any implications for the meaning of "elements of reality". You may have confused the fact that the main focus of EPR was to argue that QM is not complete rather than to define in an exhaustive manner what "reality" means. So the definition they gave which is your (2) above, is an operational definition in the context of trying to prove that QM is not complete. Therefore your claims (1), (3), (4) are not accurate representations of what EPR defined as "reality", only (2) is correct.

Dear billschnieder,

This is very helpful to me and I thank you very much for it.

I believe the EPR paper to be very important --- AND well deserving of our deep respect -- plus care in its reading.

I think I cannot better the points that you make ... just maybe add a smiley ... a small feminine touch.

But I am so pleased to see someone reading as closely as I try to do -- BUT writing better and more clearly.

Please: If you are a quantum physicist or physicist of any sort, please be very critical of my words and terminology.

1. I am sure my mathematics can take care of themselves.

2. I am also sure that my seeming direct engagement with elements of physical reality -- when coupled with my somewhat bumbling terminology -- puts most physicists off.

3. So they don't bother with the mathematics -- which I will try to put some up to day.

4. My theory stands or falls solely on the basis of my use of mathematics and probability theory, so if you will look closely at them -- that I would love.

They are not difficult -- in my opinion, sincerely.

Sorry for quick reply -- many thanks -- more soon

JenniT

 

[DrChinese]

fyi, David Bohm (1951, Quantum Theory) discusses the Einstein, Rosen, Podolsky view. Here is summary of ERP argument as understood by Bohm:

1. Every element of physical reality must have a counterpart.

2. If, before measurement the value of a physical quantity can be predicted, then there exists an element of physical reality corresponding to this physical quantity.

These two criteria rest on two assumptions:

3. The universe can be correctly analyzed in terms of distinct and separate "elements of reality"

4. Every element of physical reality must have a counterpart of a precisely defined mathematical quantity.

==

Does this summary by Bohm match your understanding of EPR ?

Despite what billschnieder says (since I disagree with almost everything he says anyway), I think this is a good quote. It covers several important lines of thought from EPR. E.g. realism and completeness.

 

[billschnieder]

Dear billschnieder,

This is very helpful to me and I thank you very much for it.

I believe the EPR paper to be very important --- AND well deserving of our deep respect -- plus care in its reading.

I think I cannot better the points that you make ... just maybe add a smiley ... a small feminine touch.

But I am so pleased to see someone reading as closely as I try to do -- BUT writing better and more clearly.

Please: If you are a quantum physicist or physicist of any sort, please be very critical of my words and terminology.

1. I am sure my mathematics can take care of themselves.

2. I am also sure that my seeming direct engagement with elements of physical reality -- when coupled with my somewhat bumbling terminology -- puts most physicists off.

3. So they don't bother with the mathematics -- which I will try to put some up to day.

4. My theory stands or falls solely on the basis of my use of mathematics and probability theory, so if you will look closely at them -- that I would love.

They are not difficult -- in my opinion, sincerely.

Sorry for quick reply -- many thanks -- more soon

JenniT

You are welcome Jenni, I am definitely following this thread closely -- not that I think you need any help with the maths or probability . Happy 4th.

 

[Gordon Watson]

LATE EDIT: THIS POST RE-SUBMITTED AS POST # 49 TO BY-PASS SERVER/TEX CLASH OVER \VBAR. \UPARROW SUBSTITUTED.

Plus some minor clarifications.

Yes, with the conditions I mentioned it's fine.

Take your time, it turns out I'll be taking a trip tomorrow and won't be back until Monday anyway.

Dear Jesse, Thank you for this very thoughtful note. Welcome back:

--------------------------------------------------A study based on Mermin, Spooky actions at a distance: Mysteries of the quantum theory. Encyclopedia Britannica, Chicago - The Great Ideas Today 1988.

NB: The singlet state (used in most EPRB-Bell tests) is invariant under rotations, not just about the line-of-flight axis.

--------------------------------------------------(1) Local realism defined:

P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;, \lambda_{k}, \lambda&#039;_{k}) = P(G\vbarH, a, \lambda_{k})*P&#039;(G&#039;\vbarH, b&#039;, \lambda&#039;_{k}).

(2) Expectation defined:

P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;) = \sum_{k=1}^N P_3(\lambda_{k}, \lambda&#039;_{k})*P(G\vbarH, a, \lambda_{k})*P&#039;(G&#039;\vbarH, b&#039;, \lambda&#039;_{k}).

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

Scene 1: Alice and Bob -- far far apart -- happily await the arrival of N twins: pairwise correlated photons -- one twin at a time to each, respectively bearing the properties \lambda_{k} and \lambda&#039;_{k}; k = 1 - N.

Alice has a dichotomic polarizer-analyzer [a/ai] which respectively signals G or R as tested photons enter the related output channel. Bob likewise has [b'/bi'] signaling respectively G' or R'. In this experiment, a is an orientation orthogonal to the line-of-flight axis, ai is orthogonal to both a and the line-of-flight axis; primes (') denote elements of reality in Bob's locale; etc.

Alice, with her elegant simplicity, supposes fundamental particles to be magic gyroscopes with perfect rotational symmetries and related binding spin trajectories. She supposes their polarizers to be devices which burn off extrinsic spin and re-orient the intrinsic spin.

Alice supposes that \lambda_{k} and \lambda&#039;_{k} are spin vectors, unconstrained as to length and orientation -- but pairwise correlated by the relevant conservation laws attending their birth. That is, for this experiment, they are pairwise equal and not otherwise.

Recognizing the rotational symmetry of the cosine function, Alice supposes the lambdas to be such that, if only she knew more about them: Without in any way disturbing a photon, she could predict with certainty the positive outcome of any spin-test upon it. Alice writes:

(3)
P(G\vbarH, a, \lambda_{k}) = (cos^2 (a, \lambda_{k})\vbar H, a, \lambda_{k}) = 0\oplus1 = 0 xor 1.
Alice, supposing that elements of physical reality mediate all her assured (certain) positive outcomes, writes:

(4)
P(G\vbarH, a, \lambda_{k}\hookrightarrow \lambda_{a}) = (cos^2 (a, \lambda_{k})\vbar H, a, \lambda_{k} \hookrightarrow \lambda_{a}) = cos^2 (a, \lambda_{a}) = 1.

(5)
P(R\vbarH, ai, \lambda_{k}\hookrightarrow \lambda_{ai}) = (cos^2 (ai, \lambda_{k})\vbar H, ai, \lambda_{k} \hookrightarrow \lambda_{ai}) = cos^2 (ai, \lambda_{ai}) = 1.

In set notation, Alice defines elements of physical reality as CFCs (counterfactual conditionals) -- binding spin trajectories associated with the perfect rotational symmetries of pristine fundamental particles:

(6)
\{\lambda_{k} \hookrightarrow \lambda_{a}\vbar \,If \,\lambda_{k} \rightarrow [a/ai] \,then \,\lambda_{k} \rightarrow \lambda_{a}\};

(7)
\{\lambda_{k} \hookrightarrow \lambda_{ai}\vbar \,If \,\lambda_{k} \rightarrow [a/ai] \,then \,\lambda_{k} \rightarrow \lambda_{ai}\}.So, in Alice's frame of reference:

(8)
P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;) = \sum_{k=1}^N P_3(\lambda_{k}, \lambda&#039;_{k})*P(G\vbarH, a, \lambda_{k})*P&#039;(G&#039;\vbarH, b&#039;, \lambda&#039;_{k}).

(9)
P_3(\lambda_{k}, \lambda&#039;_{k}) = (1/2)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})].

(10)
P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;) = \sum_{k=1}^N (1/2)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})] * (cos^2(a, \lambda_{k})\vbarH, a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k})\vbarH, b&#039;, \lambda&#039;_{k}) =

(11)
\sum_{k=1}^N (1/2)[(cos^2(a, \lambda_{k})\vbarH, a, \lambda_{k} \hookrightarrow \lambda_{a}) * (cos^2(b&#039;, \lambda&#039;_{k})\vbarH, b&#039;, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + cos^2 (a, \vbarH, a, \lambda_{k} \hookrightarrow \lambda_{ai})*(cos^2(b&#039;, \lambda&#039;_{k})\vbarH, b&#039;, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})] =

(12)
(1/2)cos^2(a&#039;, b&#039;) + 0.

QED: A result in full accord with quantum theory, quantum experiments, and local-realism. (E & O. E.)--------------------------------------------------

In Bob's frame similarly -- detail removed due earlier server/Tex interaction:

(13)
P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;) = \sum_{k=1}^N (1/2)[(\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k})\vbarH, a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k})\vbarH, b&#039;, \lambda&#039;_{k}) = (1/2) cos^2 (a, b).

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

In the frame of God and Albert -- detail removed due earlier server/Tex interaction -- but co-variant:

(14)
P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;) = \sum_{k=1}^N (1/4)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k})\vbarH, a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k})\vbarH, b&#039;, \lambda&#039;_{k}) = (1/2) cos^2 (a, b&#039;).

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

Edit 2: Alternative: For better TeX display:-

(15)

P&#039;&#039;_3(\lambda_{k}, \lambda&#039;_{k}) = (1/4)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})].

NB: Too-long line of TeX reduced by providing P&#039;&#039;_3(\lambda_{k}, \lambda&#039;_{k}) only. The reader can readily derive the conclusion as an exercise:

(16)

P&#039;&#039;(G, G&#039;\vbarH, a, b&#039;) = (1/2) cos^2 (a, b&#039;).

PS:

Note that the co-variant derivation produces (correctly, with physical significance) the separated "Alice, Bob" elements of physical reality in the cosine argument. That is: (a, b').

The earlier cosine arguments are also rigorously correct and physically significant -- respectively (a', b') for Alice's frame, (a, b) for Bob's frame -- recalling that a, ai, b, bi are just orientations in 3-space.

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

With best regards,

JenniT

 

[Gordon Watson]

You are welcome Jenni, I am definitely following this thread closely -- not that I think you need any help with the maths or probability . Happy 4th.

Thank you Bill, and same to you.

And - do you have any suggestions about the physics' terminology?

All suggestions, etc. most welcome.

JenniT

 

[Gordon Watson]

NOTE: Re Post 44 --

The server is responding intermittently and variously to my TeX coding.

History:

1. After many hours of frustration, I replaced all | with \vbar, and eliminated several long-lines of TeX code in equations.

2. The server delivered all that revised TeX incorrectly in PREVIEW -- but OK in the THREAD.

3. Checking one-hour later, the server omitted all the |-s (vbars).

4. Checking again, all the | (vbars) and their accompanying H ( - | H - as in the sample spaces of all the Probability Functions) were omitted.

5. As I write this SOME H-s are back but not the |-s (vbars) !

Any suggestions for a stable FIX, please?

Thank you,

Jenni T

EDIT -- see POST #49 for work-around using \uparrow.

 

[Gordon Watson]

DrC,

I am not clear how HUP relates to Bell's work?

Does he ever mention it?

How do you bring it into the Bell-scene please?

And with what understanding as to Bell's mathematics, if any?

Thank you,

JenniT

 

[Gordon Watson]

Despite what billschnieder says (since I disagree with almost everything he says anyway), I think this is a good quote. It covers several important lines of thought from EPR. E.g. realism and completeness.

Dear DrC,

I agree with you that this is a very good quote contributed by Salman2.

May I suggest, that from now on:

We all HONOR David BOHM by describing all so-called "tests of Bell's theorem" [in my view, they are not -- please discuss -- plus there are other descriptors] that are spin-related as

EPR-Bohm-Bell testss?

OK: How about EPRB-Bell tests?

No BOHM = No Bell?

And Bohm was not wrong, was he? ON THIS SUBJECT, nb!

Love,

JenniT

 

[Gordon Watson]

RE-SUBMISSION OF POST# 44 to minimize server/TeX clash re \vbars. \uparrow substituted.

Some small clarifications added

Yes, with the conditions I mentioned it's fine.

Take your time, it turns out I'll be taking a trip tomorrow and won't be back until Monday anyway.

Dear Jesse, Thank you for this very thoughtful note. Welcome back:

--------------------------------------------------A study based on Mermin, Spooky actions at a distance: Mysteries of the quantum theory. Encyclopedia Britannica, Chicago - The Great Ideas Today 1988.

NB: The singlet state (used in most EPRB-Bell tests) is invariant under rotations, not just about the line-of-flight axis.

--------------------------------------------------(1) Local realism defined:

P&#039;&#039;(G, G&#039; \uparrow H, a, b&#039;, \lambda_{k}, \lambda&#039;_{k}) = P(G \uparrow H, a, \lambda_{k})*P&#039;(G&#039; \uparrow H, b&#039;, \lambda&#039;_{k}).

(2) Expectation defined:

P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N P_3(\lambda_{k}, \lambda&#039;_{k})*P(G \uparrow H , a, \lambda_{k})*P&#039;(G&#039; \uparrow H , b&#039;, \lambda&#039;_{k}).

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

Scene 1: Alice and Bob -- far far apart -- happily await the arrival of N twins: pairwise correlated photons -- one twin at a time to each, respectively bearing the properties \lambda_{k} and \lambda&#039;_{k} with k = 1 - N. Note: k is a number; other subscripts -- such as a, ai, b, bi -- define spin-orientations in 3-space.

Alice has a dichotomic polarizer-analyzer [a/ai] which respectively signals G or R [often termed V or H] as tested photons enter the related output channel. Bob likewise has [b'/bi'] signaling respectively G' or R' [perhaps termed V' or H']. In this experiment, a is an orientation orthogonal to the line-of-flight axis, ai is orthogonal to both a and the line-of-flight axis; primes (') denote elements of reality in Bob's locale; etc.

Alice, with her elegant simplicity, supposes fundamental particles to be magic gyroscopes with perfect rotational symmetries and related binding spin trajectories. She supposes their polarizers to be devices which burn off extrinsic spin and re-orient the intrinsic spin.

Alice supposes that \lambda_{k} and \lambda&#039;_{k} are spin vectors, unconstrained as to length and orientation -- but pairwise correlated by the relevant conservation laws attending their birth. That is, for this experiment, they are pairwise equal and not otherwise.

Recognizing the rotational symmetry of the cosine function, Alice supposes the lambdas to be such that, if only she knew more about them: Without in any way disturbing a photon, she could predict with certainty the positive outcome of any spin-test upon it. Alice writes:

(3)
P(G \uparrow H , a, \lambda_{k}) = (cos^2 (a, \lambda_{k}) \uparrow H, a, \lambda_{k}) = 0\oplus1 = 0 xor 1.
Alice, supposing that elements of physical reality mediate all her assured (certain) positive outcomes, writes:

(4)
P(G \uparrow H , a, \lambda_{k}\hookrightarrow \lambda_{a}) = (cos^2 (a, \lambda_{k}) \uparrow H, a, \lambda_{k} \hookrightarrow \lambda_{a}) = cos^2 (a, \lambda_{a}) = 1.

(5)
P(R \uparrow H , ai, \lambda_{k}\hookrightarrow \lambda_{ai}) = (cos^2 (ai, \lambda_{k}) \uparrow H, ai, \lambda_{k} \hookrightarrow \lambda_{ai}) = cos^2 (ai, \lambda_{ai}) = 1.

In set notation, Alice defines elements of physical reality as CFCs (counterfactual conditionals) -- binding spin trajectories associated with the perfect rotational symmetries of pristine fundamental particles:

(6)
\{\lambda_{k} \hookrightarrow \lambda_{a} \uparrow \,If \,\lambda_{k} \rightarrow [a/ai] \,then \,\lambda_{k} \rightarrow \lambda_{a}\};

(7)
\{\lambda_{k} \hookrightarrow \lambda_{ai} \uparrow \,If \,\lambda_{k} \rightarrow [a/ai] \,then \,\lambda_{k} \rightarrow \lambda_{ai}\}.So, in Alice's frame of reference:

(8)
P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N P_3(\lambda_{k}, \lambda&#039;_{k})*P(G \uparrow H , a, \lambda_{k})*P&#039;(G&#039; \uparrow H , b&#039;, \lambda&#039;_{k}).

(9)
P_3(\lambda_{k}, \lambda&#039;_{k}) = (1/2)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})].

(10)
P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N (1/2)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})] * (cos^2(a, \lambda_{k}) \uparrow H , a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k}) \uparrow H , b&#039;, \lambda&#039;_{k}) =

(11)
\sum_{k=1}^N (1/2)[(cos^2(a, \lambda_{k}) \uparrow H , a, \lambda_{k} \hookrightarrow \lambda_{a}) * (cos^2(b&#039;, \lambda&#039;_{k}) \uparrow H , b&#039;, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + cos^2 (a, \lambda_{k} \uparrow H , a, \lambda_{k} \hookrightarrow \lambda_{ai})*(cos^2(b&#039;, \lambda&#039;_{k}) \uparrow H , b&#039;, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})] =

(12)
(1/2)cos^2(a&#039;, b&#039;) + 0.

QED: A result in full accord with quantum theory, quantum experiments, and local-realism. (E & O. E.)--------------------------------------------------

In Bob's frame similarly -- detail removed due earlier server/Tex interaction:

(13)
P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N (1/2)[(\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k}) \uparrow H , a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k}) \uparrow H , b&#039;, \lambda&#039;_{k}) = (1/2) cos^2 (a, b).

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

In the frame of God and Albert -- detail removed due earlier server/Tex interaction -- but co-variant:

(14)
P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N (1/4)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k}) \uparrow H , a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k}) \uparrow H , b&#039;, \lambda&#039;_{k}) = (1/2) cos^2 (a, b&#039;).

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

Edit 2: Alternative for God and Albert's frame: For better TeX display:-

(15)

P&#039;&#039;_3(\lambda_{k}, \lambda&#039;_{k}) = (1/4)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})].

NB: Too-long line of TeX reduced by providing P&#039;&#039;_3(\lambda_{k}, \lambda&#039;_{k}) only. The reader can readily derive the conclusion as an exercise:

(16)

P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = (1/2) cos^2 (a, b&#039;).

PS:

Note that the co-variant derivation produces (correctly, with physical significance) the separated "Alice, Bob" elements of physical reality in the cosine argument. That is: (a, b').

The earlier cosine arguments are also rigorously correct and physically significant -- respectively (a', b') for Alice's frame, (a, b) for Bob's frame -- recalling that a, ai, b, bi are just orientations in 3-space.

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

With best regards,

JenniT

 

[Gordon Watson]

Dear Jesse, Welcome back; and any others:

Question: How do I fix Equation (14) above -- copied below.

A long line of TeX coding does not wrap?

Is there a TeX WRAP command?

Thanks,

JenniT

--------------------------------------------------(14)
P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N (1/4)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k}) \uparrow H , a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k}) \uparrow H , b&#039;, \lambda&#039;_{k}) = (1/2) cos^2 (a, b&#039;).

 

[billschnieder]

Is there a TeX WRAP command?

I think it is double back-slash, but it does not work in the forum. You may have to use two separate tex blocks.
(14)
P&#039;&#039;(G, G&#039;|H , a, b&#039;) = \sum_{k=1}^N \frac{1}{4}[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b})
<br /> + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k}) | H , a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k}) | H , b&#039;, \lambda&#039;_{k}) = \frac{1}{2} cos^2 (a, b&#039;).

 

[Gordon Watson]

I think it is double back-slash, but it does not work in the forum. You may have to use two separate tex blocks.
(14)
P&#039;&#039;(G, G&#039;|H , a, b&#039;) = \sum_{k=1}^N \frac{1}{4}[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai}) + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{b}, \lambda_{k} \hookrightarrow \lambda_{b})
<br /> + (\lambda&#039;_{k} \hookrightarrow \lambda&#039;_{bi}, \lambda_{k} \hookrightarrow \lambda_{bi})] * (cos^2(a, \lambda_{k}) | H , a, \lambda_{k}) * (cos^2(b&#039;, \lambda&#039;_{k}) | H , b&#039;, \lambda&#039;_{k}) = \frac{1}{2} cos^2 (a, b&#039;).

Dear billschnieder,

Great, many thanks. I love it when great minds think alike. I HAD that in the early submissions but deleted it and more when the vbar problem kept occurring. &&^%$. But I am a beginner - - in TeX.

Question: Are you happy with me using P in all the equations? Rather than E in some?

I am thinking that Eqn (2) should stay with a P but change its intro to Joint probability defined?

Note: I hope I avoid the need for E anywhere here, actually; whereas Bell should have used E in his 1964 (2). But his P arguments were over A and B, not G and G', like mine?

E of course comes later when we combine the 4 P-s with their signs.

Also: Once we have one correct P, consistent with QM -- then we know the E will be correct too?

EDIT: I am not yet brave enough to re-edit Post #49 again, so will tidy it up in a new Post after some more comments are received.

Thank you again,

JenniT

 

[billschnieder]

Dear billschnieder,
Question: Are you happy with me using P in all the equations? Rather than E in some?
I am thinking that Eqn (2) should stay with a P but change its intro to Joint probability defined?

You are correct, (2) is a marginalization with respect to (λ,λ'), not an expectation. Not sure about what name is appropriate since (1) is also a joint probability.

Note: I hope I avoid the need for E anywhere here, actually; whereas Bell should have used E in his 1964 (2). But his P arguments were over A and B, not G and G', like mine?

Bells (2) is an Expectation is because he is multiplying the functions A(.), B(.) with the probability P(λ) and integrating/Summing over all λ. A(.), and B(.) are not probability distributions but two-valued functions (+1, -1). In your case, G, G' are also two valued functions (0,1). Is this correct?

E of course comes later when we combine the 4 P-s with their signs.

Are you referring to equation (9). I do not understand what you are doing in equation (9) within the square brackets. I can see it lists all the possibilities for how a specific pair of (λ, λ') can be transformed during the interaction but is not clear to me what you are adding up within the square brackets, number of case instances?

 

[Gordon Watson]

You are correct, (2) is a marginalization with respect to (λ,λ'), not an expectation. Not sure about what name is appropriate since (1) is also a joint probability.

Looks like we're out of my comfort zone and right into yours -- which is good.

Please be insistent here.

I'd like to be accurate and reflect the standards accepted by the relevant disciplines.

What about: Name (1) = Local realism defined via stochastic independence?

What about: Name (2) = Local realism defined via marginalization over local beables?

Bells (2) is an Expectation is because he is multiplying the functions A(.), B(.) with the probability P(λ) and integrating/Summing over all λ. A(.), and B(.) are not probability distributions but two-valued functions (+1, -1). In your case, G, G' are also two valued functions (0,1). Is this correct?

I think not. G, G' are discrete outcomes.

Are you referring to equation (9). I do not understand what you are doing in equation (9) within the square brackets. I can see it lists all the possibilities for how a specific pair of (λ, λ') can be transformed during the interaction but is not clear to me what you are adding up within the square brackets, number of case instances?

(9) uses JesseM's notation re P3. I'm OK with that. I personally use rho(λ, λ'), it being a normalized probability distribution (in my perhaps loose terms)? Is rho reserved for continuous distributions only?

[x], x = all the equiprobable [edit: hence the 1/2 out front] mutually-exclusive collectively-exhaustive beables. The beauty of the theory is that's all there are ... as far as Alice is concerned. Yes?

Are you OK with the use of Bell's term beables?

EDIT: In co-variant (15) there are 4 equiprobable [hence the 1/4 out front] mutually-exclusive collectively-exhaustive beables. OK?

Thanks,

JenniT

 

[billschnieder]

I'd like to be accurate and reflect the standards accepted by the relevant disciplines.

What about: Name (1) = Local realism defined via stochastic independence?

What about: Name (2) = Local realism defined via marginalization over local beables?

Not trying to be pendantic but what about
(1) = Local realism defined ...
(2) = Marginalized over local beables

I think not. G, G' are discrete outcomes.

Ah, I see.

(9) uses JesseM's notation re P3. I'm OK with that. I personally use rho(λ, λ'), it being a normalized probability distribution (in my perhaps loose terms)? Is rho reserved for continuous distributions only?

[x], x = all the equiprobable [edit: hence the 1/2 out front] mutually-exclusive collectively-exhaustive beables. The beauty of the theory is that's all there are ... as far as Alice is concerned. Yes?

Still having problems. Let's use a coin analogy, we have two coins (CC') with mutually exclusive equi-probable outcomes HH', HT', TH', TT'

I can understand why you may write P(H,H') = 1/4,
What I don't understand is why you would write it as 1/4[(H,H')+(H,T') +(T,H') + (T,T')]
If the terms within [.] represent probabilities, then each term in the square bracket is 1/4 and they necessarily sum up to 1 no? Are you trying to list all the equi-probable outcomes? If so it could be made clearer if you put that in a sentence and simply state that P(H,H') = 1/4 and be done with it.

Are you OK with the use of Bell's term beables?

I welcome the term.

 

[Gordon Watson]

Not trying to be pendantic but what about
(1) = Local realism defined ...
(2) = Marginalized over local beables

1. You will never hear me object to pedantry.

2. I usually go with Bell's Cuisine idea that these equations are consequences of local causality. Any objection or pedantic improvement? Please let me know.

Ah, I see.

Good.

Still having problems. Let's use a coin analogy, we have two coins (CC') with mutually exclusive equi-probable outcomes HH', HT', TH', TT'

OK.

I can understand why you may write P(H,H') = 1/4,
What I don't understand is why you would write it as 1/4[(H,H')+(H,T') +(T,H') + (T,T')]

It is a density; a normalised probability distribution?

If the terms within [.] represent probabilities, then each term in the square bracket is 1/4 and they necessarily sum up to 1 no?

Stop. What is in [.] are not probabilities. They are the 4 possible outcomes, each equiprobable, so put the 1/4 in front of [.]. Then you have your normalized distribution.

Perhaps I am using terms incorrectly?[/QUOTE]

Are you trying to list all the equi-probable outcomes? If so it could be made clearer if you put that in a sentence and simply state that P(H,H') = 1/4 and be done with it.

Yes, that is what I am trying to do.

NB: BUT if I do not list them all, no one seems to see the consequent picture?

Incidentally. Ignore that Alice and Bob stuff here.

I need a co-variant distribution over the whole sample space, and I need readers to see that I have such. So it is the God view that is important. Alice and Bob are but angels in this business.

I welcome the term.

Great to hear that.

Thanks,

JenniT

 

[JesseM]

Hi Jenni, sorry about the delay, I ended up being away from my apt. most of this week...

RE-SUBMISSION OF POST# 44 to minimize server/TeX clash re \vbars. \uparrow substituted.

Some small clarifications added
Dear Jesse, Thank you for this very thoughtful note. Welcome back:

--------------------------------------------------A study based on Mermin, Spooky actions at a distance: Mysteries of the quantum theory. Encyclopedia Britannica, Chicago - The Great Ideas Today 1988.

NB: The singlet state (used in most EPRB-Bell tests) is invariant under rotations, not just about the line-of-flight axis.

--------------------------------------------------(1) Local realism defined:

P&#039;&#039;(G, G&#039; \uparrow H, a, b&#039;, \lambda_{k}, \lambda&#039;_{k}) = P(G \uparrow H, a, \lambda_{k})*P&#039;(G&#039; \uparrow H, b&#039;, \lambda&#039;_{k}).

(2) Expectation defined:

P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N P_3(\lambda_{k}, \lambda&#039;_{k})*P(G \uparrow H , a, \lambda_{k})*P&#039;(G&#039; \uparrow H , b&#039;, \lambda&#039;_{k}).

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

Scene 1: Alice and Bob -- far far apart -- happily await the arrival of N twins: pairwise correlated photons -- one twin at a time to each, respectively bearing the properties \lambda_{k} and \lambda&#039;_{k} with k = 1 - N. Note: k is a number; other subscripts -- such as a, ai, b, bi -- define spin-orientations in 3-space.

Alice has a dichotomic polarizer-analyzer [a/ai] which respectively signals G or R [often termed V or H] as tested photons enter the related output channel. Bob likewise has [b'/bi'] signaling respectively G' or R' [perhaps termed V' or H']. In this experiment, a is an orientation orthogonal to the line-of-flight axis, ai is orthogonal to both a and the line-of-flight axis; primes (') denote elements of reality in Bob's locale; etc.

Alice, with her elegant simplicity, supposes fundamental particles to be magic gyroscopes with perfect rotational symmetries and related binding spin trajectories. She supposes their polarizers to be devices which burn off extrinsic spin and re-orient the intrinsic spin.

Alice supposes that \lambda_{k} and \lambda&#039;_{k} are spin vectors, unconstrained as to length and orientation -- but pairwise correlated by the relevant conservation laws attending their birth. That is, for this experiment, they are pairwise equal and not otherwise.

Recognizing the rotational symmetry of the cosine function, Alice supposes the lambdas to be such that, if only she knew more about them: Without in any way disturbing a photon, she could predict with certainty the positive outcome of any spin-test upon it. Alice writes:

(3)
P(G \uparrow H , a, \lambda_{k}) = (cos^2 (a, \lambda_{k}) \uparrow H, a, \lambda_{k}) = 0\oplus1 = 0 xor 1.

I'm not sure I understand, are you saying the angle between a and the spin vector is always going to be either 90 or 0, so that cos^2 of the angle between them is always 0 or 1? This would seem to require that the source "know in advance" what setting Alice was going to choose so it always created the particle with a spin vector parallel or orthogonal to Alice's setting, a violation of the "no-conspiracy condition" which appears in rigorous versions of Bell's proof (this condition says that P(λ) = P(λ|a), i.e. there is no correlation between hidden variables and the choice of detector setting). I suppose your condition that a and ai are orthogonal implies that the source doesn't have to know which specific choice Alice makes on each specific trial (since a lambda parallel/orthogonal to a will automatically be orthogonal/parallel to ai), but it still has to "know" the two possible angles Alice is choosing from and make sure it only emits lambdas parallel/orthogonal to them. And of course Bell's own proof does not restrict the choice of detector angles to all be parallel/orthogonal to one another, you could have an experiment with three settings a1=0, a2=60, and a3=120 for example.

Alice, supposing that elements of physical reality mediate all her assured (certain) positive outcomes, writes:

(4)
P(G \uparrow H , a, \lambda_{k}\hookrightarrow \lambda_{a}) = (cos^2 (a, \lambda_{k}) \uparrow H, a, \lambda_{k} \hookrightarrow \lambda_{a}) = cos^2 (a, \lambda_{a}) = 1.

(5)
P(R \uparrow H , ai, \lambda_{k}\hookrightarrow \lambda_{ai}) = (cos^2 (ai, \lambda_{k}) \uparrow H, ai, \lambda_{k} \hookrightarrow \lambda_{ai}) = cos^2 (ai, \lambda_{ai}) = 1.

OK, so λ_a just means a spin vector parallel to Alice's detector setting a, right? Your notation is odd--I think P(G \uparrow H , a, \lambda_{k}\hookrightarrow \lambda_{a}) means "the probability of result G given experiment H, detector setting a, and a value of k such that \lambda_k = \lambda_a"...if that's right, I think it would be more in line with standard notation conventions to write it as P(G | H, a, \lambda_k = \lambda_a )

In set notation, Alice defines elements of physical reality as CFCs (counterfactual conditionals) -- binding spin trajectories associated with the perfect rotational symmetries of pristine fundamental particles:

(6)
\{\lambda_{k} \hookrightarrow \lambda_{a} \uparrow \,If \,\lambda_{k} \rightarrow [a/ai] \,then \,\lambda_{k} \rightarrow \lambda_{a}\};

(7)
\{\lambda_{k} \hookrightarrow \lambda_{ai} \uparrow \,If \,\lambda_{k} \rightarrow [a/ai] \,then \,\lambda_{k} \rightarrow \lambda_{ai}\}.

I don't follow this. What "set notation" do you mean? What does the uparrow represent? Earlier you seemed to just replace the | denoting conditional probability with an uparrow, but these equations don't seem to be expressing conditional probabilities. Can you explain in words what these equations are saying?

So, in Alice's frame of reference:

(8)
P&#039;&#039;(G, G&#039; \uparrow H , a, b&#039;) = \sum_{k=1}^N P_3(\lambda_{k}, \lambda&#039;_{k})*P(G \uparrow H , a, \lambda_{k})*P&#039;(G&#039; \uparrow H , b&#039;, \lambda&#039;_{k}).

(9)
P_3(\lambda_{k}, \lambda&#039;_{k}) = (1/2)[(\lambda_{k} \hookrightarrow \lambda_{a}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{a}) + (\lambda_{k} \hookrightarrow \lambda_{ai}, \lambda&#039;_{k} \hookrightarrow \lambda&#039;_{ai})].

Is the right side of equation 9 supposed to involve probabilities? If so I think it would make more sense to write it as P_3(\lambda_{k}, \lambda&#039;_{k}) = (1/2)[P(\lambda_{k} = \lambda_{a}, \lambda&#039;_{k} = \lambda&#039;_{a}) + P(\lambda_{k} = \lambda_{ai}, \lambda&#039;_{k} = \lambda&#039;_{ai})]...is that OK? Even so the equation is a little unclear, does the 1/2 just mean you assume P(\lambda_{k} = \lambda_{a}, \lambda&#039;_{k} = \lambda&#039;_{a}) = 1/2 and P(\lambda_{k} = \lambda_{ai}, \lambda&#039;_{k} = \lambda&#039;_{ai}) = 1/2, i.e. these are the only two possible ways the source can emit spin vectors and both ways are equally probable? Then the sum of the two would be 1, and you'd have to multiply that by 1/2 to get P_3(\lambda_{k}, \lambda&#039;_{k}) for any specific value of k (k=a or k=ai). If that's the idea, I think it would be a lot less convoluted to just say that k can only take one of two values, k=a or k=ai, (which could perhaps be represented by the numerals 1 and 2, so the sum would just be \sum_{k=1}^2) and that either way P_3(\lambda_{k}, \lambda&#039;_{k}) = 1/2. In that case we could go from equation (8) to a simplified version of (10):

P&#039;&#039;(G,G&#039;|H,a,b&#039;) = \sum_{k=1}^2 (1/2)*(cos^2(a,\lambda_k)|H,a,\lambda_k)*(cos^2(b&#039;,\lambda_k)|H,b,\lambda_k)

Let me know what you think of my comments/questions/suggested changes to notation so far, then we can proceed.

 

[Gordon Watson]

Hi Jenni, sorry about the delay, I ended up being away from my apt. most of this week...

I'm not sure I understand, are you saying the angle between a and the spin vector is always going to be either 90 or 0, so that cos^2 of the angle between them is always 0 or 1? This would seem to require that the source "know in advance" what setting Alice was going to choose so it always created the particle with a spin vector parallel or orthogonal to Alice's setting, a violation of the "no-conspiracy condition" which appears in rigorous versions of Bell's proof (this condition says that P(λ) = P(λ|a), i.e. there is no correlation between hidden variables and the choice of detector setting). I suppose your condition that a and ai are orthogonal implies that the source doesn't have to know which specific choice Alice makes on each specific trial (since a lambda parallel/orthogonal to a will automatically be orthogonal/parallel to ai), but it still has to "know" the two possible angles Alice is choosing from and make sure it only emits lambdas parallel/orthogonal to them. And of course Bell's own proof does not restrict the choice of detector angles to all be parallel/orthogonal to one another, you could have an experiment with three settings a1=0, a2=60, and a3=120 for example.

The angle between a and the spin vector is always going to be either 90 or 0, so that cos^2 of the angle between them is always 0 or 1. That is what polarizers do: They orient the randomly arriving (and so randomly specified k) particles in just that way.

So I believe that your questions should move to the random specifications? Yes?

Especially as you fix my notation below with an impossibility. Which is not a criticism of you at all -- just an excellent way to show what I cannot say. AND MY notation sure could do with some help -- to make it more "standard" to the disciplines of maths and physics.

OK, so λ_a just means a spin vector parallel to Alice's detector setting a, right? Your notation is odd--I think P(G \uparrow H , a, \lambda_{k}\hookrightarrow \lambda_{a}) means "the probability of result G given experiment H, detector setting a, and a value of k such that \lambda_k = \lambda_a"...if that's right, I think it would be more in line with standard notation conventions to write it as P(G | H, a, \lambda_k = \lambda_a )

Oh NO, goodness NO!

Could I get away with that!?

Really, that would be too easy, surely? Yes?

As I see it: This is the standard trick in QM -- as I say, as I see it.

You are assigning to an unmeasured photon a property that has probability zero prior to any measurement. It at the same time being a property that the subject photons will certainly have after the measurement interaction?

You have here the crux of my case, I think?

I don't follow this. What "set notation" do you mean? What does the uparrow represent? Earlier you seemed to just replace the | denoting conditional probability with an uparrow, but these equations don't seem to be expressing conditional probabilities. Can you explain in words what these equations are saying?

See note given earlier, below. Is that OK? How would you word it?

Is the right side of equation 9 supposed to involve probabilities? If so I think it would make more sense to write it as P_3(\lambda_{k}, \lambda&#039;_{k}) = (1/2)[P(\lambda_{k} = \lambda_{a}, \lambda&#039;_{k} = \lambda&#039;_{a}) + P(\lambda_{k} = \lambda_{ai}, \lambda&#039;_{k} = \lambda&#039;_{ai})]...is that OK? Even so the equation is a little unclear, does the 1/2 just mean you assume P(\lambda_{k} = \lambda_{a}, \lambda&#039;_{k} = \lambda&#039;_{a}) = 1/2 and P(\lambda_{k} = \lambda_{ai}, \lambda&#039;_{k} = \lambda&#039;_{ai}) = 1/2, i.e. these are the only two possible ways the source can emit spin vectors and both ways are equally probable? Then the sum of the two would be 1, and you'd have to multiply that by 1/2 to get P_3(\lambda_{k}, \lambda&#039;_{k}) for any specific value of k (k=a or k=ai). If that's the idea, I think it would be a lot less convoluted to just say that k can only take one of two values, k=a or k=ai, (which could perhaps be represented by the numerals 1 and 2, so the sum would just be \sum_{k=1}^2) and that either way P_3(\lambda_{k}, \lambda&#039;_{k}) = 1/2. In that case we could go from equation (8) to a simplified version of (10):

P&#039;&#039;(G,G&#039;|H,a,b&#039;) = \sum_{k=1}^2 (1/2)*(cos^2(a,\lambda_k)|H,a,\lambda_k)*(cos^2(b&#039;,\lambda_k)|H,b,\lambda_k)

NOW please see here, and note please: Any specific value of k (k=a or k=ai) has probability zero before a test and equiprobable certainty after a test. This simple dichotomy being the basis of our 2 lovely equivalence classes.

How would you put this FACT in your words and notation, please?

Also: Avoiding that zero-probability clash -- how about this?

What if we said something like this?

''Whereas before k was a number -- k = 1- N -- we now let k = 1 be the proxy for all the twins in equivalence class 1, and k = 2 be the proxy all the twins in equivalence class 2.''

How might you word that to be compatible with your specialist discipline?

Let me know what you think of my comments/questions/suggested changes to notation so far, then we can proceed.

Dear Jesse, so glad to learn that you are back home, safe and sound, and in good form -- from what I see here.

This is just a short reply to say "many thanks" again. And to knock off some rough edges.

1. To the last para: OK, will do.

2. The \uparrow was used to by-pass my big-time server clashes with too many \vbars. It is to be read as \vbar throughout (despite how much I like it as it is).

3. The set notation piece is simply saying that we are defining (just two will do) disjoint sets by giving the condition that the elements of each set must satisfy.

4. I think I see again the lovely simplification that billschnieder pointed out. Thanks to you too for that.

Pardon my caution in not replying too quickly: I need to be sure that absolutely nothing is lost or presumed when we simplify stuff.

PS: Some EDITS inserted above by me might help you comment progressively? I hope they do not confuse you?

More soon,

XXOOXXOO

Jenni

 

[JesseM]

The angle between a and the spin vector is always going to be either 90 or 0, so that cos^2 of the angle between them is always 0 or 1. That is what polarizers do: They orient the randomly arriving (and so randomly specified k) particles in just that way.

So do you mean the λ_k in cos^2(a, λ_k) refers to the angle of the particle's spin vector after it's been reoriented by the polarizer (when the angle between λ_k and a is always guaranteed to be 0 or 90), not the angle of the spin vector given to the particle when it's first sent out by the source, before any such reorientation? Is cos^2 supposed to be the probability it will pass through the polarizer (so the polarizer is imagined to first reorient, then allow to pass through or not) or is it the probability of something else happening after passing through it, and if so what?

Also, do the polarizers reorient the spin vectors in a deterministic way, so that if both polarizers are at the same angle and both spin vectors start out at the same k before passing through the polarizer, they are guaranteed to both end up parallel or both end up orthogonal, with no chance of one ending up parallel to the polarizer and one ending up orthogonal? If so we can say that at the moment the two particles first begin their journey from the source, their initial spin vectors should give them a predetermined answer to what direction they would end up if they met a polarizer at any given angle, right? For example, we might have a situation where one particle's initial spin vector (the spin vector assigned to it by the source, prior to encountering the polarizers) is such that it if it encountered a polarizer at a=30 it would be predetermined to end up parallel to it, predetermined to come out orthogonal a polarizer at a=60, and predetermined to come out parallel to a polarizer at a=120. Is this how you see things, or do you not agree that the angle of the initial spin vector plus the angle of the polarizer should deterministically generate the angle of the spin vector after being reoriented by the polarizer?

OK, so λ_a just means a spin vector parallel to Alice's detector setting a, right? Your notation is odd--I think P(G \uparrow H , a, \lambda_{k}\hookrightarrow \lambda_{a}) means "the probability of result G given experiment H, detector setting a, and a value of k such that \lambda_k = \lambda_a"...if that's right, I think it would be more in line with standard notation conventions to write it as P(G | H, a, \lambda_k = \lambda_a )

Oh NO, goodness NO!

"No" to what? Is my English summary of the meaning of your equation as "the probability of result G given experiment H, detector setting a, and a value of k such that \lambda_k = \lambda_a" incorrect? And given your point about "reorientation" here which I hadn't caught before, does λ_k in this equation represent the angle of the spin vector after it's been reoriented by the spin vector (when it's guaranteed to be parallel or orthogonal to a) or to the spin vector prior to reorientation? Also, does λ_a mean a spin vector parallel to a, with λ_ai representing a spin vector orthogonal to a?

You are assigning to an unmeasured photon a property that has probability zero prior to any measurement.

What property are you talking about?

I'll wait for your answers before continuing...

 

[Gordon Watson]

So do you mean the λ_k in cos^2(a, λ_k) refers to the angle of the particle's spin vector after it's been reoriented by the polarizer (when the angle between λ_k and a is always guaranteed to be 0 or 90), not the angle of the spin vector given to the particle when it's first sent out by the source, before any such reorientation?

As I see it, we are dealing with Probability Functions. They map a subset of the Sample Space to [0, 1].

λ_k is the angle of the spin vector given to the particle when it is first sent out by the source.

If that were not the case, it would be written λ_X, or something.

Any change in the sample space relating to λ_k will be reflected in the mapping.

It is my understanding that polarizers deterministically re-orient the spin.

So the Sample Space changes ... so the mapping changes correspondingly.

Is cos^2 supposed to be the probability it will pass through the polarizer (so the polarizer is imagined to first reorient, then allow to pass through or not) or is it the probability of something else happening after passing through it, and if so what?

Doesn't it have to be the first?

Also, do the polarizers reorient the spin vectors in a deterministic way, so that if both polarizers are at the same angle and both spin vectors start out at the same k before passing through the polarizer, they are guaranteed to both end up parallel or both end up orthogonal, with no chance of one ending up parallel to the polarizer and one ending up orthogonal?

This is Mermin's baby. In my view the answer here is Yes.

If it is something else, like a different singlet state, we just change the correlation and proceed.

If so we can say that at the moment the two particles first begin their journey from the source, their initial spin vectors should give them a predetermined answer to what direction they would end up if they met a polarizer at any given angle, right?

That is correct.

Is that what is required?

For example, we might have a situation where one particle's initial spin vector (the spin vector assigned to it by the source, prior to encountering the polarizers) is such that it if it encountered a polarizer at a=30 it would be predetermined to end up parallel to it, predetermined to come out orthogonal a polarizer at a=60, and predetermined to come out parallel to a polarizer at a=120. Is this how you see things,...

Yes.

IS THAT HOW IT SHOULD BE? Sorry for caps - hit wrong key.

... or do you not agree that the angle of the initial spin vector plus the angle of the polarizer should deterministically generate the angle of the spin vector after being reoriented by the polarizer?

Does that question make sense?

The angle of the initial spin vector plus the angle of the polarizer deterministically generate the NEW angle of the spin vector WITHIN THE the polarizer. THE PHOTON THEN ENTERS THE RELEVANT ANALYZER CHANNEL.

"No" to what? Is my English summary of the meaning of your equation as "the probability of result G given experiment H, detector setting a, and a value of k such that \lambda_k = \lambda_a" incorrect?

Yes. This property ... \lambda_k = \lambda_a ... has probability zero before the test!

Do you see that?

And given your point about "reorientation" here which I hadn't caught before, does λ_k in this equation represent the angle of the spin vector after it's been reoriented by the spin vector [SIC] POLARIZER Yes?

(when it's guaranteed to be parallel or orthogonal to a) or to the spin vector prior to reorientation?

No -- see above re Sample Space. You can't change λ_k in the trig argument except via the SS.

Also, does λ_a mean a spin vector parallel to a, with λ_ai representing a spin vector orthogonal to a?

Yes.

What property are you talking about?

This property \lambda_k = \lambda_a has probability zero before the test!

I'll wait for your answers before continuing...

Regards,

JenniT