What's wrong with this local realistic counter-example to Bell's theorem?

[JesseM]

The Pab++ example can be checked right now. It is the worked example in the PDF.

Yes, I checked it, and it was wrong.

This is rushed, but are you looking at the right table. The PDF has the correct result.

Yes, I was looking just at the PDF. There you write that

Pab++=P3+P4=(2Sab + Cac.Sbc + Sac.Cbc)/6=(2Sab + 2Pab++)/6=Sab/2

But with the specific examples I gave of a=240,b=120,c=0 it is trivial to see that they are not equal:

(2Sab + Cac.Sbc + Sac.Cbc)/6=0.3125

Sab/2=0.375

So these are not equal to one another, and if we use Pab++=0.3125 neither of these is equal to (2Sab + 2Pab++)/6=0.3541666... If we use Pab++=0.375 then it is true that (2Sab + 2Pab++)/6 = Pab++, but neither of these is consistent with the earlier equation in the PDF saying that Pab++=P3+P4=(2Sab + Cac.Sbc + Sac.Cbc)/6.

The substitutions that you query are given in post #33.

There you will see: Pab++ = P3 + P4 = (Cac.Sbc + Sac.Cbc)/2.

In post #33 you continue to use the incomprehensible language of "bi-angles", but in any case it's clear that the equation Pab++ = P3 + P4 = (Cac.Sbc + Sac.Cbc)/2 from post #33 is inconsistent with the PDF's equation of Pab++=P3+P4=(2Sab + Cac.Sbc + Sac.Cbc)/6, because with a=240,b=120,c=0 we have:

(Cac.Sbc + Sac.Cbc)/2 = (0.25*0.75 + 0.75*0.25)/2 = 0.1875

whereas

(2Sab + Cac.Sbc + Sac.Cbc)/6=(2*0.75 + 0.25*0.75 + 0.75*0.25)/6 = 1.875/6 = 0.3125

and neither of these are equal to

Sab/2 = 0.375

Please check this numerical example yourself before responding, you'll see that what I say is correct.

 

[JesseM]

Refer back to DrC's post. (#2?, iirc)

If the proposed LR model doesn't produce an LR dataset (and therefore satisfy BI), then it isn't an LR model. Period.

And if it does produce an LR dataset then it can't agree with qm (and, so far and presumably, with experimental results). Period.

There's nothing else to consider. LR models of entanglement are impossible. Period.

What does this tell us of the underlying reality? Nothing. Period.

Thomas, you seem to be arguing as if "local realist" is defined to mean you must have a dataset of predetermined values, but this isn't true. As I said in post #20 to Avodyne:

Not all Bell inequalities assume that the hidden variables totally determine what the response to any given angle will be, see the CHSH inequality for example where the angles a,a' used on one side may be different from the angles b,b' on the other in which case there'd be no combination of settings where knowledge of the result on one side gives you total certainty about the result on the other. I like to think of the definition of local realism this way:

1. The complete set of physical facts about any region of spacetime can be broken down into a set of local facts about the value of variables at each point in that regions (like the value of the electric and magnetic field vectors at each point in classical electromagnetism)

2. The local facts about any given point P in spacetime are only causally influenced by facts about points in the past light cone of P, meaning if you already know the complete information about all points in some spacelike cross-section of the past light cone, additional knowledge about points at a spacelike separation from P cannot alter your prediction about what happens at P itself (your prediction may be a probabilistic one if the laws of physics are non-deterministic).

From this it follows that if two experimenters making measurements at a spacelike separation are guaranteed to get the same result when they choose the same angle, despite the fact that they pick which angle to use at random, then the results for each possible angle must have already predetermined by facts in their past light cone (like facts about the hidden variables associated with each particle) at some time T after the particles had been emitted but before they picked their angles. But this is just a derived consequence of the notion of local realism above, not a definition of local realism.

Do you agree that no theory with characteristics 1 and 2 above (regardless of whether it involves datasets of predetermined results, and regardless of whether the local variables are hidden or measurable as in classical electromagnetism which does satisfy 1 and 2) could reproduce the statistics predicted by QM?

 

[ThomasT]

Thomas, you seem to be arguing as if "local realist" is defined to mean you must have a dataset of predetermined values, but this isn't true.

I agree, see my post #64 in this thread. However, the LR dataset (ie., agreement with BI) is a necessary condition that any proposed LR model has to satisfy.

That's the beauty of DrC's LR dataset requirement and the math of Bell's theorem. It cuts through, ie. obviates, what you're going through here. I just didn't appreciate or understand it before because I was lost in the trees so to speak. If Jennit's model of entanglement agrees with qm and violates Bell's inequality, then, via the definition of an LR model a la Bell's archetype and your 1 and 2 (which entails that an LR model will satisfy Bell's inequality), it can't be an LR model of entanglement. The same holds for Christian's Clifford algebra C-space model, or Unnikrishnan's static phase relation (iirc) model or any other model that reproduces qm results and violates Bell's inequality.

Do you agree that no theory with characteristics 1 and 2 above (regardless of whether it involves datasets of predetermined results, and regardless of whether the local variables are hidden or measurable as in classical electromagnetism which does satisfy 1 and 2) could reproduce the statistics predicted by QM?

Yes. The point is that a necessary condition for a theory to be called LR is that it has to produce datasets that don't agree with the qm predicted datasets -- ie., that it satisfies Bell's inequality.

And now I think I should probably delete my second, rather terse, post in this thread. I just had to savor the irony (recalling the countless interchanges that we've had on this subject). Or should I leave it? Your call.

 

[JesseM]

Yes. The point is that a necessary condition for a theory to be called LR is that it has to produce datasets that don't agree with the qm predicted datasets -- ie., that it satisfies Bell's inequality.

OK good, I just wanted to make sure you understood that although satisfying Bell's inequality is a necessary consequence of the basic definition of LR, it isn't part of the definition itself (otherwise Bell would have no need for a "proof" that local realism implies the inequality is satisfied!) So I would disagree with your earlier statement "What does this tell us of the underlying reality? Nothing. Period." It tells us that QM is incompatible with any theory satisfying point 1) and 2) from my post to Avodyne, I would say that's a strong negative result ruling out a broad class of theories about the "underlying reality", although perhaps you just meant that it doesn't give us a positive answer to what the underlying reality actually is.

And now I think I should probably delete my second, rather terse, post in this thread. I just had to savor the irony (recalling the countless interchanges that we've had on this subject). Or should I leave it? Your call.

I don't think there's a need to delete it, perhaps the resulting discussion could clarify some things for people reading the thread.

 

[ThomasT]

OK good, I just wanted to make sure you understood that although satisfying Bell's inequality is a necessary consequence of the basic definition of LR, it isn't part of the definition itself (otherwise Bell would have no need for a "proof" that local realism implies the inequality is satisfied!)

Yes, I get this -- it just took a while.

So I would disagree with your earlier statement "What does this tell us of the underlying reality? Nothing. Period." It tells us that QM is incompatible with any theory satisfying point 1) and 2) from my post to Avodyne, I would say that's a strong negative result ruling out a broad class of theories about the "underlying reality", although perhaps you just meant that it doesn't give us a positive answer to what the underlying reality actually is.

I mean that it doesn't tell us whether what's happening in the underlying reality is due to exclusively local interactions and transmissions or not. So, it doesn't contradict the general assumptions of locality and the existence of local hidden variables. It just tells us that explicitly LR formulations of entanglement must produce a reduced range of statistical predictions, and are therefore ruled out.

Why this is so is what Hess, Michielsen, and De Raedt are talking about in their recent paper on the subject.

 

[JesseM]

I mean that it doesn't tell us whether what's happening in the underlying reality is due to exclusively local interactions and transmissions or not.

How do you figure? Wouldn't any theory about the underlying reality which involves "exclusively local interactions and transmissions" satisfy 1) and 2) in my description of local realism?

So, it doesn't contradict the general assumptions of locality and the existence of local hidden variables.

I don't understand why you would distinguish "the general assumptions of locality and the existence of local hidden variables" from "local realism" as described by 1) and 2). Are you imagining there could be a local theory involving local hidden variables which violated either 1) or 2) (or both)? If so which would it violate?

 

[Dmitry67]

Sorry for jumping in,
but (on a psycological level) why "die hard local realists" (c) Dr. Chinese
never consider just using MWI to obtain the result they want?

 

[Gordon Watson]

Clearly, it can't. (there's a typo for Pac++ and Pbc++ where the values are 0.0732, but I guess this is a typo).

I didn't check your pdf in detail, but in your pdf, you (correctly) define:

Pab++ = P3 + P4

You work this out to something (I didn't check the goniometric algebra), but it equals P3 + P4.

You didn't work out the rest, but if it is done correctly, you should also have:

Pcb++ = P3 + P7

and

Pac++ = P2 + P4.

It is a pity that you didn't work it out.

Now, I can assume that the numerical value of P3 + P7 doesn't change between when you write it as P3 + P7, and when you work out the goniometric algebra
(otherwise you made a mistake in your algebra, right ?)

Now, give me please the NUMERICAL VALUES for a vertically, c 45 degrees towards the window, and b horizontal towards the window (so 90 degrees), for all 8 values P1, ... P8 and then for your calculation of Pab++ , Pac++ and Pcb++.

Because it should be obvious that if all your values P1...P8 are positive numbers, and if Pab++ = P3 + P4 and so on as you claim (correctly), that you CANNOT obtain numerically Pab++ = 0.25 and Pac++ = Pcb++ = 0.073...

So in order to show you this, you should work out, for the given angles, the numbers P1 up to P8, and then Pab++ = P3 + P4 and also according to your algebra Sab/2 and Pcb++ and Pac++.

Thanks for your patience and blunder-identification (which I've corrected via an edit). Time-pressure is the excuse; trying to hold up my end of this thread.

That blunder is the message for now:

I've made a big mistake in trying to do my bit here and keep my end of the thread moving while snowed-under with other critical commitments: because I am lapping up the teaching and I'm keen to get to the end.

I even delayed my departure, that's how I'm here, to get some further calcs and answers out.

So: Thanks again; I shall return; your specific questions addressed; quick calcs and answers to follow later -- now to be happily reviewed more carefully first.

The hope for us all, if I may add some hope, is that the arguments will be resolved via maths, not words. That I am happy with; that I learn best from. XO

 

[ThomasT]

How do you figure? Wouldn't any theory about the underlying reality which involves "exclusively local interactions and transmissions" satisfy 1) and 2) in my description of local realism?

Yes, but that doesn't imply that the constraints on statistical predictions associated with the formal LR requirements are due to an underlying nonlocality. I mean, you can interpret it that way, but I don't think you have to. Isn't it at least possible that the effective determiner of BI violations has to do with some more mundane conflict between the LR formalism and the experimental design and preparation of Bell tests than with the existence of nonlocal signals or the nonexistence of local hidden variables?

I don't understand why you would distinguish "the general assumptions of locality and the existence of local hidden variables" from "local realism" as described by 1) and 2).

As DrC's signature quotes from Korzybski, "The map is not the territory." (Did I get that right?)

Are you imagining there could be a local theory involving local hidden variables which violated either 1) or 2) (or both)?

No. I'm just holding out for a more parsimonious explanation for BI violation (that I fully understand) than that it's due to nonlocality. Who knows, maybe I'll eventually think that Bell's theorem proves nonlocality, but right now that doesn't seem likely.

Edit: Apologies to JenniT for somewhat off topic posts. I thought you were going to be away for a while and I wanted to express and clarify my thinking on this. Anyway, I'll post no more in this thread so that you might continue your presentation and argument.

 

[JesseM]

Yes, but that doesn't imply that the constraints on statistical predictions associated with the formal LR requirements are due to an underlying nonlocality.

I didn't use the word "nonlocality", I just said that a theory of the type described by 1) and 2) was ruled out. As you know from previous discussions there are certain "loopholes", for example I was implicitly assuming in 1) that there is a unique set of physical facts about each point in spacetime and so a unique result to any specific measurement, if you drop this and imagine multiple parallel versions of a measurement occurring in the same region of spacetime as in the MWI, then you may be able to explain the quantum statistics without violating locality. Similarly if you imagine the experimenter's choice of what detector setting to use on each trial is not actually uncorrelated with the local variables associated with the particle immediately after emission, so that you have a sort of retrocausal effect where the particle "anticipates" what the future detector setting will be, then one might argue that this would be compatible with locality as well (though some might argue that retrocausal influences don't count as "local").

As DrC's signature quotes from Korzybski, "The map is not the territory." (Did I get that right?)

I don't see how that statement is applicable here. Would you describe statements 1) and 2) about the nature of physics as "map" or "territory"? Whichever you'd choose, I don't see why you'd say that "the general assumptions of locality and the existence of local hidden variables" was any different. And logically, if you agree that "the general assumptions of locality and the existence of local hidden variables" (or 'exclusively local interactions and transmissions") would be false if 1) and 2) were false, then if QM is incompatible with 1) and 2) that shows it's also incompatible with "the general assumptions of locality and the existence of local hidden variables" regardless of what you call "map" and what you call "territory".

 

[ThomasT]

1. The complete set of physical facts about any region of spacetime can be broken down into a set of local facts about the value of variables at each point in that regions (like the value of the electric and magnetic field vectors at each point in classical electromagnetism)

2. The local facts about any given point P in spacetime are only causally influenced by facts about points in the past light cone of P, meaning if you already know the complete information about all points in some spacelike cross-section of the past light cone, additional knowledge about points at a spacelike separation from P cannot alter your prediction about what happens at P itself (your prediction may be a probabilistic one if the laws of physics are non-deterministic).

I don't understand why you would distinguish "the general assumptions of locality and the existence of local hidden variables" from "local realism" as described by 1) and 2).

My current thinking is that 1) is obviated by the experimental design of Bell tests. Accurately predicting entanglement correlations simply doesn't require breaking things down into sets of local facts. It's because of the inclusion of a local hidden variable lambda in the formulation that the range of statistical predictions is reduced.

2) entails a formalism that contradicts both parameter and outcome dependence. But outcome dependence doesn't contradict locality, thus facilitating a more parsimonious explanation for why BIs are violated than the existence of underlying nonlocal transmissions.

 

[ThomasT]

... I just said that a theory of the type described by 1) and 2) was ruled out.

And we're in agreement on that, which represents progress in my understanding. Maybe we should just leave it at that for the time being and I'll stay out of the thread.

 

[JesseM]

My current thinking is that 1) is obviated by the experimental design of Bell tests.

Purely by the design of the tests, or by the resulting statistics? If you just think that the design itself is enough to obviate 1) you are misunderstanding something, you could certainly do the same sort of tests in a universe where classical electromagnetism was exactly correct and 1) and 2) would both be correct there, you just wouldn't get any statistics that violated Bell inequalities in this case. Keep in mind that 1) doesn't forbid you from talking about "facts" that involve an extended region of spacetime, it just says that these facts must be possible to deduce as a function of all the local facts in that region. For example, in classical electromagnetism we can talk about the magnetic flux through an extended 2D surface of arbitrary size, this is not itself a local quantity, but the total flux is simply a function of all the local magnetic vectors at each point on the surface, that's the sort of thing I meant when I said in 1) that all physical facts "can be broken down into a set of local facts". Similarly in certain Bell inequalities one considers the expectation values for the product of the two results (each one represented as either +1 or -1), obviously this product is not itself a local fact, but it's a trivial function of the two local facts about the result each experimenter got.

2) entails a formalism that contradicts both parameter and outcome dependence. But outcome dependence doesn't contradict locality, thus facilitating a more parsimonious explanation for why BIs are violated than the existence of underlying nonlocal transmissions.

Not sure what you mean by "parameter and outcome dependence", can you be more specific? Again, 1) and 2) are definitely true of classical electromagnetism, if you think they aren't (for any conceivable experiment in a universe where classical electromagnetism was exactly true) then you are misunderstanding something (do you think any possible experiment in such a universe would display parameter and/or outcome dependence?)

 

[Gordon Watson]

I'm back. And happy to be so.

Apologies for my absence; me recognizing my part in this tread. And many thanks to all participants.

Absences will continue sporadically. But there's no need for me to signal them: In that I am committed to answer every question here, you can rest assured that I have not quit ... until I say so.

If a final post sinks my model, then you can expect me to acknowledge that, and move happily on.

I have no problem correcting my wrong beliefs and moving on to new (and better) ones.

I trust the above is a satisfactory part-reply to TomT. Note that my maths is elementary, and supports what I would have thought to be a reasonable LR position for you. More re DrC's position soon, that might interest you.

To Avodyne, OK, and fair enough. But I hope to go beyond semantics and hand-waving.

To Dmitry67, thanks for your post! I'm very happy to spar with physicists who have neither knuckle-dusters, nor FTL, not NL, in their gloves. If I survive this warm-up bout, I might be ready for this one: What's wrong with this One World Interpretation of MWI? Until then, thanks again.

More soon, with a focus on technical issues raised by DrC, JesseM, vanesch.

XO-s and thanks again to all,

JenniT

 

[Gordon Watson]

The criticism is the same for all such: it isn't realistic! ("So any inference to a third side will be misleading. ")

If it is, simply provide a dataset for us to look at. 0, 120, 240 degrees is always a good combo to supply. We will see if the QM predictions hold.

I will repeat my main objection again: it's not realistic if you do not provide values for measurements which cannot be performed. That is the definition of "realistic".

The "confusion" issue is: to the extent anyone agrees with you, we are simply talking about the usual approach to Bell or a closely related equivalent variation. To the extent you assert you have provided a LR counter-example, we keep explaining that actually you have violated the requirement of L locality or R realism despite your words. You cannot just wave your hands and say you have accomplished this without pointing us to some new revelation. I see nothing novel in your approach at all, and it seems to follow your arguments presented in other threads.

Where's the beef? It would really be nice if you would show us something new to discuss rather than just say "I'm right unless you show me where am I wrong".

I have tidied up the presentation, in the attached PDF, in the hope of minimising confusion: and would welcome your comments on it; plus:

Do most of the Tables give results for experiments that cannot be performed? (I think that they do.)

Can you point to any hand-waving in the PDF please? (I am keen to delete any such.)

Could we discuss a protocol for studying your 0, 120, 240 example; along the lines of Figure 1 in the PDF, and related commentary thereunder? (I would be happy to derive the subsequent results.)

PS: For JesseM and vanesch: I am working on replies to your welcome technical queries; please don't despair.

And thank you, as always, DrC.

 

[Gordon Watson]

I have tidied up the presentation, in the attached PDF, in the hope of minimising confusion: and would welcome your comments on it; plus:

Do most of the Tables give results for experiments that cannot be performed? (I think that they do.)

Can you point to any hand-waving in the PDF please? (I am keen to delete any such.)

Could we discuss a protocol for studying your 0, 120, 240 example; along the lines of Figure 1 in the PDF, and related commentary thereunder? (I would be happy to derive the subsequent results.)

PS: For JesseM and vanesch: I am working on replies to your welcome technical queries; please don't despair.

And thank you, as always, DrC.

Re the above post by me:

A. I suggest that we refer to the PDF (attached to the above post) as PDF2.

B. CORRECTIONS to PDF2:

1. Equation (A3a); the last term should read [Sbc + P(bc++|a)]/3.

2. In the second line of the Headings to Tables A1.a, A2.b, A3.c: delete the 2.

 

[Gordon Watson]

I highlighted what we need: Pac++, Pab++ and Pcb++=Pbc--

Pac++ is what is measured on monday, and equals 0.073... in agreement with your numbers
Pab++ is what is measured on tuesday, and equals 0.25. Your number gives 0.125
Pcb++=Pbc-- is what is measured on thursday, and equals 0.073 in agreement with your numbers.

You will NEVER be able to get Pab++ equal to 0.25 (you actually have 0.125), simply because it can't be larger than (Pac++) + (Pbc--) which equals 0.146...
Note that indeed, your number (0.125) is, as it can't be otherwise, smaller than 0.146.
Simply because this 0.146 is made up of your 4 positive numbers P2 + P3 + P4 + P7 as you give it yourself, and Pab++ is equal to only P3+P4 (your 0.125). You ADD to your 0.125 still your P2 and P7 to obtain 0.146, so it has to be smaller (as indeed it is).

Now, QM predicts not 0.125, but rather 0.25. It is bigger. So it can't come from numbers P1...P8 in this way.

There's nothing more to say about this.

Please refer to PDF2, attached at this post

In PDF2 (see above), I have clarified the notation by including the conditioning space in every Probability function. That conditioning, now explicit, was implicit (as you will see) in the example that you cite. The RO was given as c, and the output statement was explicit in referring Pab to = the average over the bi-angle. [2Pab corrected to Pab.]

That is, as also in PDF2: Pab(++|c) = P3 + P4 = (Cac.Sbc + Sac.Cbc)/2.

So we now examine your relations with the implicit conditioning space now explicit (as in PDF2, Table A3.c):

A: P(ac++|c) is what is measured on Monday, and equals 0.073 in agreement with my numbers.

B: P(ab++|c) [SIC] is what is measured on Tuesday, and equals 0.25. [I agree with 0.25. BUT you say that my number gives 0.125: Your statement is incorrect -- as shown below.]

C: P(cb++|c) =P(bc--|c) is what is measured on Thursday, and equals 0.073 in agreement with my numbers.

So our disagreement is at B only ... and, I believe, readily turned to agreement:

Please note that what is measured on Tuesday is P(ab++|ab) or P(ab++|a) or P(ab++|b)! With my model, you have three choices as to how you define it. And from PDF2, all equal Sab/2 = 0.25. In full agreement with YOUR calculation.

The one choice that you cannot make (with my model) is this: That on Tuesday we measured P(ab++|c).

Reason: Orientation c was nowhere evident in Tuesday's test.

Tuesday's test used orientation a, orientation b, and angle ab; the model can work with any of these. BUT orientation c CANNOT appear in the conditioning space for Tuesday's test.

With this correction, which I trust you understand and accept, there is nothing more to say beyond this: We agree with the QM numbers that apply to the subject tests.

In that I said that my model correctly delivered all the QM results, this agreement was to be expected.

So let me now see if we can reach agreement re bi-angles: According to the model, Tuesdays bi-angles are 0 and 90, and the experimenters chose to measure over the 90 value: No problem whatsoever for the model. But note: One bi-angle value yields (S0)/2 = 0. The other bi-angle value yields (S45)/2 = 0.25. The average of these results is 0.125.

That is the origin of that 0.125 number; which is not the number applicable to the actual measurement made on Tuesday. The model gives BOTH numbers, and both correctly: The measured result is 0.25 (in full agreement with QM), and the average over the bi-angle 0.125.

Where we still differ is in the numbers that you invoke re the (supposedly) related Bell-inequality. But as PDF2 states: In agreement with QM (so in agreement between you and me, I'm sure), my model will disagree on numbers to do with BT.

So I will now move to reply to the post where you gave such numbers and, from memory, related them to an impossibility that is unrelated to my model. [The model does not fail when it comes to BT-based impossibilities. Rather, it shows that they cannot be rationally constructed from within. This is shown in PDF2, equation (?). Yes, equation(?), foot of page 4.]

I seek to show you that the L*R model again agrees with QM; here re the futility of any attempt to construct BT from within L*R.

In closing: I very much appreciate your attention to detail, and your engagement with the model. At the end of the day, I expect us both to agree on all the QM numbers. AND on QM's position that BT cannot be constructed from within QM.

I go on to say that BT cannot be constructed from within L*R ("advanced local realism"). So I see that that is where our discussion will head; e.g., is L*R truly L + R. In that MWI beats BT too, as I (preliminarily) understand it, it might boil down to us uniting L*R and MWI -- who knows --

With many thanks, as always; more soon.

 

[JesseM]

You still seem to be talking about "bi-angles" in your pdf, and the diagram shows an experimental setup where, if "a" is an orientation pointing vertically, then there are two possible choices for the direction of "b" and two possible choices for the direction of "c". Please understand that this is not the experimental setup envisioned by Bell or the one that's used in actual Bell test experiments. In a Bell test experiment, the experimenters have pre-agreed on only three possible orientations for the Stern-Gerlach devices or polarizers--imagine that there's a clock face on the wall in front of the SG device used by one of the experimenters, and that experimenter must arrange his device so that the North end of the North-South axis of his device is either pointing in the same direction as 12 o'clock, 2 o'clock, or 4 o'clock, no other orientations are permitted (meanwhile the other experimenter is only allowed to pick orientations which would match up with the same readings in a mirror image of the first experimenter's clock). It is in this specific experiment that Bell said it would be impossible for a local realist theory to violate the Bell inequality, not the alternate setup you seem to be imagining where if "a" points at 12 o'clock "b" could either point at 2 o'clock or 12-2 = 8 o'clock. So, I hope you will take this into consideration and avoid all reference to "bi-angles" in any future response to me.

 

[Gordon Watson]

You still seem to be talking about "bi-angles" in your pdf, and the diagram shows an experimental setup where, if "a" is an orientation pointing vertically, then there are two possible choices for the direction of "b" and two possible choices for the direction of "c". Please understand that this is not the experimental setup envisioned by Bell or the one that's used in actual Bell test experiments. In a Bell test experiment, the experimenters have pre-agreed on only three possible orientations for the Stern-Gerlach devices or polarizers--imagine that there's a clock face on the wall in front of the SG device used by one of the experimenters, and that experimenter must arrange his device so that the North end of the North-South axis of his device is either pointing in the same direction as 12 o'clock, 2 o'clock, or 4 o'clock, no other orientations are permitted (meanwhile the other experimenter is only allowed to pick orientations which would match up with the same readings in a mirror image of the first experimenter's clock). It is in this specific experiment that Bell said it would be impossible for a local realist theory to violate the Bell inequality, not the alternate setup you seem to be imagining where if "a" points at 12 o'clock "b" could either point at 2 o'clock or 12-2 = 8 o'clock. So, I hope you will take this into consideration and avoid all reference to "bi-angles" in any future response to me.

OK; fair enough. But please see my most recent reply to vanesch re the conditioning spaces (of the probability functions), that must be applied to any real experiment carried out over 3 orientations. Also, equations (A0a) - (A0c), exemplifying the derivation of the QM results, make no mention of those angles. Do you see a problem with this equation set?

1. Without reference to any angles, other than those specifically tested (as requested), Table 2 in PDF2 provides all the testable probabilities; all in accord with QM. Do we agree on that? Does this Table, with supporting equations, answer some of your earlier questions re what it is that Table 1 delivers.

2. Note that the alternative "test-arrangements" were given in PDF2 as a way of illustrating what it is that L*R does. Do you believe that the allowance of these additional tests would somehow be the way that L*R breaches BT? (They are not.) Remember that Alice and Bob can move on to any RO, and L*R will still deliver the correct outcome distributions; see Table 2.

3. Moreover, from those correct distributions, L*R delivers precise values for any test set-up. As shown in PDF2, equation (?), there is no basis for a BI in L*R, anymore than there is such a basis within QM.

Many thanks.

 

[JesseM]

OK; fair enough. But please see my most recent reply to vanesch re the conditioning spaces (of the probability functions), that must be applied to any real experiment carried out over 3 orientations.

Since I don't understand the meaning of your "reference orientation" I don't understand what type of "conditioning space" you are using, as in your response to vanesch you say "The RO was given as c" and then condition all your probabilities on c. Normally a probability like Pab(++) would be conditioned on the fact that Alice chose angle "a", Bob chose angle "b", while the specific values of "a", "b" and "c" would be assumed as part of the conditions of the experiment. Indeed that seems to be what you do in Table 2 of the pdf when you write conditional probabilities like P(ab++|ab), but elsewhere in the pdf (and in your response to vanesch) you condition on other things besides the choice of two detector settings on a given trial, I don't understand the reason for that.

Also, equations (A0a) - (A0c), exemplifying the derivation of the QM results, make no mention of those angles. Do you see a problem with this equation set?

Where are equations (A0a) - (A0c)? If they're in the PDF, what page?

1. Without reference to any angles, other than those specifically tested (as requested), Table 2 in PDF2 provides all the testable probabilities; all in accord with QM. Do we agree on that? Does this Table, with supporting equations, answer some of your earlier questions re what it is that Table 1 delivers.

No, I already showed the math for getting Table 2 from the probabilities in Table 1 doesn't work in [post=3159151]post 71[/post] which I hope you will review and respond to. If P(ab++|ab)=P3+P4, then according to Table 1 this will be:

[Sab.Cac + Sab.Sbc + Cac.Sbc]/6 + [Sab.Sac + Sab.Cbc + Sac.Cbc]/6 =
[Sab*(Cac + Sac) + Sab*(Sbc + Cbc) + Cac.Sbc + Sac.Cbc]/6 =
[2Sab + Cac.Sbc + Sac.Cbc]/6

And as I showed in post #71, for the angles a=240,b=120,c=0, this would be equal to 0.3125. But Table 2 claims that P(ab++|ab)=Sab/2, and for these angles Sab/2=0.375. So, the equations in Table 1 are inconsistent with Table 2, assuming you accept equations such as P(ab++|ab)=P3+P4.

2. Note that the alternative "test-arrangements" were given in PDF2 as a way of illustrating what it is that L*R does. Do you believe that the allowance of these additional tests would somehow be the way that L*R breaches BT? (They are not.)

Your question makes no sense to me. The BT deal with a specific type of experiment, how would could the "allowance of additional tests" involving a totally different type of experiment be a way of breaching a theorem which doesn't address the second type of experiment at all? This is kind of like asking whether the "allowance" of an experiment on the breeding habits of Bengal tigers would "be the way that L*R breaches BT".

 

[Gordon Watson]

Since I don't understand the meaning of your "reference orientation" I don't understand what type of "conditioning space" you are using, as in your response to vanesch you say "The RO was given as c" and then condition all your probabilities on c. Normally a probability like Pab(++) would be conditioned on the fact that Alice chose angle "a", Bob chose angle "b", while the specific values of "a", "b" and "c" would be assumed as part of the conditions of the experiment. Indeed that seems to be what you do in Table 2 of the pdf when you write conditional probabilities like P(ab++|ab), but elsewhere in the pdf (and in your response to vanesch) you condition on other things besides the choice of two detector settings on a given trial, I don't understand the reason for that.

A reference orientation is any pre-agreed orientation of a detector in a Bell-test experiment, the experimenters (Alice and Bob) having here pre-agreed to three such orientations. (Thank you.)

The conditioning space in P(X|C) is C; as usual. And, as usual, the boundary conditions on the experiment are implied. So, as I believe is customary, only special conditions are included specifically. Since P(ab++|ab) differs from P(ab++|c), the model includes both. P(ab++|ab) is the probability of ab++ when correlated across the angle ab. P(ab++|c) arises from the use of frames of reference in the model, as in my reply to vanesch. I'll address this when I revise; in that I will be removing the Bell-test objection that you raise with regard to multiple tests.

I honestly expected no resistance to ADDITIONAL tests, over all 3 reference orientations. And thought it (for present purposes), better, less novel, than introducing the local-realistic basis for L*R. Will fix.

Where are equations (A0a) - (A0c)? If they're in the PDF, what page?

PDF2, Page 7.

No, I already showed the math for getting Table 2 from the probabilities in Table 1 doesn't work in [post=3159151]post 71[/post] which I hope you will review and respond to. If P(ab++|ab)=P3+P4, then according to Table 1 this will be:

[Sab.Cac + Sab.Sbc + Cac.Sbc]/6 + [Sab.Sac + Sab.Cbc + Sac.Cbc]/6 =
[Sab*(Cac + Sac) + Sab*(Sbc + Cbc) + Cac.Sbc + Sac.Cbc]/6 =
[2Sab + Cac.Sbc + Sac.Cbc]/6

And as I showed in post #71, for the angles a=240,b=120,c=0, this would be equal to 0.3125. But Table 2 claims that P(ab++|ab)=Sab/2, and for these angles Sab/2=0.375. So, the equations in Table 1 are inconsistent with Table 2, assuming you accept equations such as P(ab++|ab)=P3+P4.

PDF2 was written to correct the hurried mess that was the first PDF, with its short-cuts; including short-cutting the conditioning space on any P; noting that all P are conditional to me, in that some conditions are inevitably implied or explicit. So, with apologies, PDF2 now spells out every calculation. Which will bring you to those darned angles.

Your question makes no sense to me. The BT deal with a specific type of experiment, how would could the "allowance of additional tests" involving a totally different type of experiment be a way of breaching a theorem which doesn't address the second type of experiment at all? This is kind of like asking whether the "allowance" of an experiment on the breeding habits of Bengal tigers would "be the way that L*R breaches BT".

I thought that, when the L*R tests are conducted over every Bell-test detector setting (as they are; see Tables A1-A3), all we would have (and agree upon) is a more complete set of Bell-tests. I did not expect an objection to MORE tests; any component of which is a component of a Bell-test.

As for tigers: I thought that I was breeding Bell-tests.

So, for me, the question remains: Given that Table 2, PDF2, will remain unchanged, and since it applies to any Bell-test that you might nominate, how will a Bell Inequality be constructed?

To take a common example: How will P(ab++|ab) be yoked to P(ac++|ac) and P(bc++|bc)? The model being as one with QM in this respect?

It seems to me that these considerations will eventually move us to enquire: Is the model truly local and realistic?

That is: Local and realistic in line with Einstein's ideas and expectations? Me believing that he was not happy with the EPR elements of reality (too naive, imho), me understanding that he makes no mention of them in his work?

 

[JesseM]

A reference orientation is any pre-agreed orientation of a detector in a Bell-test experiment, the experimenters (Alice and Bob) having here pre-agreed to three such orientations. (Thank you.)

The conditioning space in P(X|C) is C; as usual. And, as usual, the boundary conditions on the experiment are implied. So, as I believe is customary, only special conditions are included specifically.

Yes, that's how conditional probability normally works, you don't include conditions which are present in all trials (like facts about the experimental setup which don't change), you only include conditions which can vary from one trial to another. But then I don't understand why you write:

Since P(ab++|ab) differs from P(ab++|c), the model includes both.

P(ab++|ab) means we are looking at only the subset of trials where Alice chose angle "a" and Bob chose angle "b", correct? But then what does P(ab++|c) mean? If c is supposed to be the "reference orientation", you just said the "reference orientation" was "pre-agreed", so it shouldn't change from one trial to another. Does c here not represent the choice of which orientation to call the "reference orientation" but rather the actual detector setting chosen by Alice or Bob on a trial? If so, who chose c, Alice or Bob? Or both? You really need to explain your notation more when it departs from the standard notation.

P(ab++|ab) is the probability of ab++ when correlated across the angle ab.

"correlated across angle ab" is an odd way of phrasing it, do you mean the same thing as I meant, i.e. the probability they both get result "+" in the subset of trials where Alice chose to set her SG device at the orientation "a" and Bob chose to set his SG device at the orientation "b"?

P(ab++|c) arises from the use of frames of reference in the model, as in my reply to vanesch.

There is no need to consider multiple "frames of reference", we can just use a single physical standard for labeling the three possible orientations, as I said in this comment:

In a Bell test experiment, the experimenters have pre-agreed on only three possible orientations for the Stern-Gerlach devices or polarizers--imagine that there's a clock face on the wall in front of the SG device used by one of the experimenters, and that experimenter must arrange his device so that the North end of the North-South axis of his device is either pointing in the same direction as 12 o'clock, 2 o'clock, or 4 o'clock, no other orientations are permitted (meanwhile the other experimenter is only allowed to pick orientations which would match up with the same readings in a mirror image of the first experimenter's clock)

I'll address this when I revise; in that I will be removing the Bell-test objection that you raise with regard to multiple tests.

I honestly expected no resistance to ADDITIONAL tests, over all 3 reference orientations.

I have no idea what you mean by "over all 3 reference orientations", is this something to do with your bizarre notions about "bi-angles" and multiple "coordinate systems"? You seem to want to play a weird shell game where you try to "win" by using different labels for the same physical orientations on different trials, but surely you understand that a mere re-labeling is not going to change the actual results of any physical experiment. Why not accept the standard practice in physics of using a single scheme for labeling angles of physical objects, rather than developing some completely weird and idiosyncratic labeling scheme that makes everything far more complicated for no apparent reason?

Where are equations (A0a) - (A0c)? If they're in the PDF, what page?

Page 7; it's safe; no tigers lurking.

OK, as noted above I don't know what terms like P(ab++|a) even mean, and if it's something to do with changing how you label angles from one trial to another, I don't really want to know. Unless you are making the totally crackpot argument that proving Bell wrong requires this sort of relabeling (in which case I really have no interest in trying to reason with you), please just adopt the standard practice of picking a single way to label angles and sticking with it through all trials. Note that I already asked you to do this in two separate posts...in post #25 I said:

Look, if you want to talk about angles there's no need for some convoluted notion of defining them relative to one another and picking one as a "reference angle", just do what is always done when talking about angles in physics, and define them relative to some fixed coordinate system! You could have a long straight rod stretching from one experimenter to the other whose position never changes and which is taken to define the x-axis of your coordinate system, and then the angle of the polarizer could just be defined as the angle relative to the rod, and then if you started the polarizer out parallel to the rod you could just see how many degrees you have to rotate it counterclockwise before it reaches the desired orientation, and call that the "angle" of the desired orientation. In this case every orientation would have a well defined angle, like a=70, b=30 and c=10, and then a difference between two angles like ac could just be defined as one minus the other, so ac=a-c while ca=c-a and so forth. In this case it's clear that ac=ab+bc is true since (a-c)=(a-b)+(b-c), while ac=ab-bc is false since (a-c)=(a-b)-(b-c)=a-2b+c which doesn't work. Given my example angles above you can see that ac=70-10=60, ab=70-30=40, and bc=30-10=20, so clearly ac=ab+bc does work since 60=40+20, but ac=ab-bc doesn't since 60 is not equal to 40-20.

I really hope your entire argument doesn't reduce to an incoherent notation for labeling angles...if not, then please just phrase your argument in terms of the standard type of coordinate-based angular notation I describe above.

And in post #29 I said:

I would like you to use the standard type of notation for angles, where individual angles are defined relative to some fixed coordinate angles and differences between two angles are defined in some fixed way, like ab=a-b. If you think the terminology of "bi-angles" still makes sense in this context, then please explain clearly what you mean, hopefully using a numerical example where we have definite angles for a,b,c and can thus calculate any angles like ab and ac.

...

Your notion of "focusing" on 2 angles or "reference angles" are similarly incomprehensible to me, I'm just talking about angles in the standard way that physicists always talk about angles, defining them relative to some fixed coordinate system, see post #25. As I requested there, I would like you to start using this sort of standard definition of angles as well, if your argument really revolves around saying there is something fundamentally flawed about defining angles relative to a fixed coordinate system and that we must use your incomprehensible alternative definitions, then your argument really is hopelessly crackpot and I am not interested in continuing.

Will you agree to this, and not refer me to any arguments or equations involving changing definitions of which orientation is at an angle of 0 and what the angles of the other two orientations are?

PDF2 was written to correct the hurried mess that was the first PDF, with its short-cuts; including short-cutting the conditioning space on any P; noting that all P are conditional to me, in that some conditions are inevitably implied or explicit. So, with apologies, PDF2 now spells out every calculation. Which will bring you to those darned angles.

PDF2 involves a lot of incomprehensible notation and terminology such as "bi-angles" and "P(ab++|c)". In your reply to me, assuming we are using a fixed coordinate system where the angles assigned to each orientation are a=240,b=120,c=0, can you please show what values you would calculate for P3, P4 and P(ab++|ab) given these angles?

I thought that, when the L*R tests are conducted over every Bell-test detector setting (as they are; see Tables A1-A3), all we would have (and agree upon) is a more complete set of Bell-tests. I did not expect an objection to MORE tests; any component of which is a component of a Bell-test.

This is all totally incomprehensible to me, as I did not object to "more tests", I objected to the notion (which seemed implicit in your "bi-angles" terminology and in Fig. 1) that in a single series of trials the physical meaning of a given label like "b" might be different on some trials than others (on some it might mean the SG device was aligned with 2 o'clock, on some it might mean it was aligned with 10 o'clock...again, that's what Fig. 1 seems to show). Bell inequalities are only meant to apply to a series of trials where the experimenters were picking between a set of three physical orientations which are labeled in a consistent way, of course you could first do a series of trials #1 where the orientations were a=240,b=120,c=0 and a series of trials #2 where the orientations were a=30,b=20,c=0, but each series would have a separate Bell inequality it would be expected to satisfy under local realism, Bell wouldn't say that local realism demands that when you combine the data from both series that the combined dataset must still satisfy a Bell inequality.

So, for me, the question remains: Given that Table 2, PDF2, will remain unchanged, and since it applies to any Bell-test that you might nominate, how will a Bell Inequality be constructed?

Table 2 simply gives the standard QM probabilities, what I don't believe is that you can derive Table 2 from any table like Table 1 that gives specific values for P1-P8 and is thus compatible with local realism. If you disagree, please address my specific example of a=240,b=120,c=0, and tell me what values you would get for P3 and P4 given these angles. From Table 1 it seems you should get P3=0.9375/6=0.15625 and P4=0.15625, so that if P(ab++|ab)=P3+P4 this would imply P(ab++|ab)=0.15625 + 0.15625 = 0.3125, but Table 2 says that P(ab++|ab)=Sab/2 which for these angles is equal to 0.375.

To take a common example: How will P(ab++|ab) be yoked to P(ac++|ac) and P(bc++|bc)? The model being as one with QM in this respect?

Um, the whole point of the argument is that none of us believe you can come up with a local realist model that gives probabilities for P1-P8 which is "as one with QM". Do you really not understand the really really basic point that all Bell inequalities are supposed to be claims about what must be true under local realism, not about what is predicted by the QM probabilities?

It seems to me that these considerations will eventually move us to enquire: Is the model truly local and realistic?

That is: Local and realistic in line with Einstein's ideas and expectations? Me believing that he was not happy with the EPR elements of reality (too naive, imho), me understanding that he makes no mention of them in his work?

I have my doubts that you understand either "the EPR elements of reality" or the standard notion of "local realism"--perhaps you could explain what aspects you find "naive" so we could see if you are addressing what these ideas actually mean or just some strawman version.

 

[Gordon Watson]

I have my doubts that you understand either "the EPR elements of reality" or the standard notion of "local realism"--perhaps you could explain what aspects you find "naive" so we could see if you are addressing what these ideas actually mean or just some strawman version.

Excuse me for chopping up you last post here. All points will be answered. I just want to separate out some non-mathematical issues first.

I call "naive" any local realism that does not allow that a measurement may perturb the measured system. My view makes me wary of the way some interpret the EPR paper; i.e., when they conclude that if particle 1 is measured to be spin-UP, then particle 2 is spin-UP prior to its measurement. (In my view, a measurement of one reveals an equivalence class for the other -- which is quite a different statement -- and one which I trust will not side-track us here from moving to a focus on my maths.)

By "locality", I follow Einstein (1949): " ... the real factual situation of the system S2 is independent of what is done with system S1, which is spatially separated from the former."

With "realism", I follow Clauser and Shimony (1978): "Realism is a philosophical view, according to which external reality is assumed to exist and have definite properties, whether or not they are observed by someone." This means that I talk about trajectories and total angular momenta before they are measured.

Does this remove your concern in this area?

 

[JesseM]

Excuse me for chopping up you last post here. All points will be answered. I just want to separate out some non-mathematical issues first.

I call "naive" any local realism that does not allow that a measurement may perturb the measured system. My view makes me wary of the way some interpret the EPR paper; i.e., when they conclude that if particle 1 is measured to be spin-UP, then particle 2 is spin-UP prior to its measurement. (In my view, a measurement of one reveals an equivalence class for the other -- which is quite a different statement -- and one which I trust will not side-track us here from moving to a focus on my maths.)

I agree with you, there's no need to take this "naive" view that measurements are simply revealing properties of the particle that were exactly the same before measurement. I haven't looked at the EPR paper lately so I can't say for sure that they avoid this naive view, but from my reading of Bell's own work I'm confident that his version of local realism did not take such a naive view. In post #20 I gave my summary of how I understand "local realism", which I think matches Bell's conception:

1. The complete set of physical facts about any region of spacetime can be broken down into a set of local facts about the value of variables at each point in that regions (like the value of the electric and magnetic field vectors at each point in classical electromagnetism)

2. The local facts about any given point P in spacetime are only causally influenced by facts about points in the past light cone of P, meaning if you already know the complete information about all points in some spacelike cross-section of the past light cone, additional knowledge about points at a spacelike separation from P cannot alter your prediction about what happens at P itself (your prediction may be a probabilistic one if the laws of physics are non-deterministic).

Then in post #83 I offered the following clarification to ThomasT about the meaning of 1):

Keep in mind that 1) doesn't forbid you from talking about "facts" that involve an extended region of spacetime, it just says that these facts must be possible to deduce as a function of all the local facts in that region. For example, in classical electromagnetism we can talk about the magnetic flux through an extended 2D surface of arbitrary size, this is not itself a local quantity, but the total flux is simply a function of all the local magnetic vectors at each point on the surface, that's the sort of thing I meant when I said in 1) that all physical facts "can be broken down into a set of local facts". Similarly in certain Bell inequalities one considers the expectation values for the product of the two results (each one represented as either +1 or -1), obviously this product is not itself a local fact, but it's a trivial function of the two local facts about the result each experimenter got.

The "local facts" referred to in 1) might or might not be facts about the values of measurable quantities like position and spin, there is no requirement that they correspond in any such direct way to measurable quantities (and if they do, there is certainly no requirement that the value of this quantity at a point on the particle's worldline immediately before measurement is the same as its value at a point immediately after measurement). All that's required in my clarification to ThomasT is that a fact about a measurement performed in some finite region of spacetime, like "we measured the particle and got result spin-up", would be in principle deducible from the complete set of all local facts about points in spacetime within that finite region, there should be no "irreducibly nonlocal" facts in the universe which cannot even in principle be deduced from the complete set of local facts.

By "locality", I follow Einstein (1949): " ... the real factual situation of the system S2 is independent of what is done with system S1, which is spatially separated from the former."

And would you agree that the "real factual situation" cannot be "irreducibly nonlocal" as I defined it above? That whatever the "real factual situation" about some well-defined region of spacetime, it must ultimately boil down to a collection of real factual situations about each point in spacetime within that region?

Also, Einstein's statement requires some clarification, it's not just that they are "spatially separated" in the ordinary sense of being at different positions in space, but that we are talking about two regions of spacetime with a spacelike separation, so that we are not talking about the factual situation about one system in a region of spacetime that's in the past light cone of the region of spacetime of the other system. After all, if S1 was in the past light cone of S2 then facts about S1 could have a causal influence on facts about S2, so they wouldn't necessarily be statistically "independent". That's why I defined point 2) in terms of light cones above.

With "realism", I follow Clauser and Shimony (1978): "Realism is a philosophical view, according to which external reality is assumed to exist and have definite properties, whether or not they are observed by someone." This means that I talk about trajectories and total angular momenta before they are measured.

As long as you agree with the above point that this external reality can in principle be boiled down to a collection of local facts about each point in spacetime, and the other point that there must be a spacelike separation between two sets of local facts for them to be considered truly independent, then I think your definition shouldn't be any different from mine above. But please tell me if you have any objections to (or questions about) my definition of local realism (which I'm pretty sure matches up with Bell's notion), if you think there's any way in which it differs from your own.

 

[Gordon Watson]

I agree with you, there's no need to take this "naive" view that measurements are simply revealing properties of the particle that were exactly the same before measurement. I haven't looked at the EPR paper lately so I can't say for sure that they avoid this naive view, but from my reading of Bell's own work I'm confident that his version of local realism did not take such a naive view. In post #20 I gave my summary of how I understand "local realism", which I think matches Bell's conception:

1. The complete set of physical facts about any region of spacetime can be broken down into a set of local facts about the value of variables at each point in that regions (like the value of the electric and magnetic field vectors at each point in classical electromagnetism)

2. The local facts about any given point P in spacetime are only causally influenced by facts about points in the past light cone of P, meaning if you already know the complete information about all points in some spacelike cross-section of the past light cone, additional knowledge about points at a spacelike separation from P cannot alter your prediction about what happens at P itself (your prediction may be a probabilistic one if the laws of physics are non-deterministic).

Then in post #83 I offered the following clarification to ThomasT about the meaning of 1):

The "local facts" referred to in 1) might or might not be facts about the values of measurable quantities like position and spin, there is no requirement that they correspond in any such direct way to measurable quantities (and if they do, there is certainly no requirement that the value of this quantity at a point on the particle's worldline immediately before measurement is the same as its value at a point immediately after measurement). All that's required in my clarification to ThomasT is that a fact about a measurement performed in some finite region of spacetime, like "we measured the particle and got result spin-up", would be in principle deducible from the complete set of all local facts about points in spacetime within that finite region, there should be no "irreducibly nonlocal" facts in the universe which cannot even in principle be deduced from the complete set of local facts.

And would you agree that the "real factual situation" cannot be "irreducibly nonlocal" as I defined it above? That whatever the "real factual situation" about some well-defined region of spacetime, it must ultimately boil down to a collection of real factual situations about each point in spacetime within that region?

Also, Einstein's statement requires some clarification, it's not just that they are "spatially separated" in the ordinary sense of being at different positions in space, but that we are talking about two regions of spacetime with a spacelike separation, so that we are not talking about the factual situation about one system in a region of spacetime that's in the past light cone of the region of spacetime of the other system. After all, if S1 was in the past light cone of S2 then facts about S1 could have a causal influence on facts about S2, so they wouldn't necessarily be statistically "independent". That's why I defined point 2) in terms of light cones above.

As long as you agree with the above point that this external reality can in principle be boiled down to a collection of local facts about each point in spacetime, and the other point that there must be a spacelike separation between two sets of local facts for them to be considered truly independent, then I think your definition shouldn't be any different from mine above. But please tell me if you have any objections to (or questions about) my definition of local realism (which I'm pretty sure matches up with Bell's notion), if you think there's any way in which it differs from your own.

Jesse, this is fantastic stuff for me, and I want to do all that I can to keep it coming -- hopefully to the point of a full consensus between us. And I continue to marvel at your "fluency" (efficiency) across various threads. (I need to lift my game in that area.)

But here's my problem: I neatly began to "itemize" your prior post, to begin developing the clearest possible answers (having nothing to hide, and keen to learn). Then BANG, another set of "itemizations" required, and I haven't finished with the first post!

NOW, that's my problem, so please do not change your style. Keep pumping the info and questions out; and chase me up on any point not clear or missed.

I just want to be clear why I will sometimes appear to be guarded in my answers; why I may appear to be over-cautious in some replies: I want to reassure you that my ideas are grounded in a great deal of good sense, so that that you will move ahead with more facts and probing questions.

I will cover them all, as the thread progresses. And, for me, seeing the consequences that might be (improperly) associated with some of my early answers, I can clarify such points as I go.

SO, to the point, on your latest post:

I cannot immediately see where we disagree; or might disagree.

I certainly accept that added "space-like" requirement without question. And if I'd found an old paper of mine (as I now have), I would simply have cut and pasted this (re photons, hence the nu):

"That is, following Einstein: The real factual situation of a system v [nu] is independent of what is done to system v' that is space-like separated from it," (Einstein 1949).

My immediate question relates to this: there should be no "irreducibly nonlocal" facts in the universe which cannot even in principle be deduced from the complete set of local facts.

It is not an impediment to any analysis of my model, but it seems to me that it could be worded more clearly? Could you put it another way? Some clarifying punctuation, maybe?

Many thanks, as always; and henceforth to be understood.

 

[Gordon Watson]

THIS IS THE LAST POST BY JenniT

This will be my last post at PF.

As some of you know, from private communications, I am the tentative alter ego of a keen PF supporter.

That supporter, my boyfriend, struggles with his writing. So I represent his attempt to develop a suitable "social-networking" style of expression and correspondence

That same boyfriend (it's our 4th anniversary on April 9) has proposed that we live together. And I've accepted.

We've been allocating duties, jobs, etc, via coin-tosses (and no funny-business ...) ever since, with these results:

Here's where I won: He moves into my flat next Saturday. I am to have lots of babies. I am to find affordable land in a happy valley by the sea to continue our research into developing heirloom fruit and vegetables that grow like weeds. I am to sort out his research in this area.

Here's where he lost: He is to be the bread-winner. He is to make lots of babies, starting next Saturday. (Yes, truly; we start then!) He, poor boy, is to sort out my messy interests in physics.

In closing: I would like to sincerely thank every PF participant that has contributed to my knowledge and experience and learning here; especially those who might have thought that we were squabbling; or me too cheeky ... sometimes. I apologize for such shortcomings, though I do believe that I learn best when it's fun.

I thank Greg for creating PF, and for his efficient administration. (I have advised him of my departure.)

I thank DrC and ThomasT for their informative inputs, and for their mutual goings-on; me seemingly at odds with them both.

I thank vanesch for sharing his knowledge, and the way he brings mathematics into his answers.

I especially thank JesseM for his patience, diligence and all-round competence and knowledge. He has helped me very much! "One day with a great teacher beats a thousand days studying solo."

I have learned a great deal at PF, and will watch from afar, now and then, with great interest.

Ciao, for now,

XOXOXOXOX

JenniT

 

[Gordon Watson]

This is the first post by Gordon Watson.

Symbolically turning over a new leaf, as a reminder to myself, I would like to assure readers of this thread re two things:

1. All questions will be answered in due course.

2. I will certainly acknowledge that penultimate post, should there be such, that sinks the model definitively.

With best regards,

Gordon Watson

 

[ThomasT]

The "local facts" referred to in 1) might or might not be facts about the values of measurable quantities like position and spin, there is no requirement that they correspond in any such direct way to measurable quantities (and if they do, there is certainly no requirement that the value of this quantity at a point on the particle's worldline immediately before measurement is the same as its value at a point immediately after measurement). All that's required in my clarification to ThomasT is that a fact about a measurement performed in some finite region of spacetime, like "we measured the particle and got result spin-up", would be in principle deducible from the complete set of all local facts about points in spacetime within that finite region, there should be no "irreducibly nonlocal" facts in the universe which cannot even in principle be deduced from the complete set of local facts.

And would you agree that the "real factual situation" cannot be "irreducibly nonlocal" as I defined it above? That whatever the "real factual situation" about some well-defined region of spacetime, it must ultimately boil down to a collection of real factual situations about each point in spacetime within that region?

so, Einstein's statement requires some clarification, it's not just that they are "spatially separated" in the ordinary sense of being at different positions in space, but that we are talking about two regions of spacetime with a spacelike separation, so that we are not talking about the factual situation about one system in a region of spacetime that's in the past light cone of the region of spacetime of the other system. After all, if S1 was in the past light cone of S2 then facts about S1 could have a causal influence on facts about S2, so they wouldn't necessarily be statistically "independent". That's why I defined point 2) in terms of light cones above.

Wrt my current understanding, Bell codifies your 1) via his 'continuous λ', and codifies your 2) via his locality condition.

Regarding 1):

The inclusion of a λ variable is necessary for the realism part of a Local Realistic (LR) model to be clearly evident in the model. That is, if it isn't explicitly realistic in that it includes explicit reference to a local hidden variable (λ), then by what criterion might the model be said to be realistic? I think the answer is that there is no other way to do it than by inclusion of λ. Without λ, then a proposed LR model lacks R. (Unless Gordon Watson is proposing some sort of non-Bell-like R, and can demonstrate why it should be considered R.) And it has therefore failed the first test in determining whether it's an LR model of entanglement.

The reason I said in an earlier post that 1) is not necessary wrt effective modelling of entanglement (whether everything is actually evolving according to local causality or not -- and of course we have no way of ascertaining that) is that we know that λ is irrelevant wrt determining coincidental photon flux. λ determines individual photon flux. A continuous λ allows us to trace the production of a relationship between λ_a and λ_b back to the emission process and codifies the assumption that this relationship had a local, common cause (eg., wrt Aspect, two photons are entangled, via conservation of angular momentum, via their emission by the same atom during the same atomic transition).

One hypothesis is that the inclusion of λ in a model of entanglement skews the range of statistical results that will be predicted by such a model independent of whether the evolution of the underlying reality excludes nonlocal transmissions.

But if no λ, then the model isn't LR. And if it includes R, then either L is encoded via R or an additional locality condition is required. Sort of a Catch-22 for LR diehards if the above hypothesis is correct -- since it's untestable.Regarding 2):

The inclusion of a λ variable still doesn't codify the causal independence of S1 and S2. For that we need some sort of locality condition. Bell's reduces to an expression of both causal and simple statistical independence. Hence, we have no way of knowing whether BIs are effectively experimentally violated via one or the other.Regarding Gordon Watson's (GW) proposed LR model:

What I would like to see is a clear exposition and explanation of GW's LR ansatz. If, at that stage, it's ascertained to be nonrealistic, then whether it reproduces qm results is moot, because it wouldn't be an LR model.

That is, I still don't understand exactly how GW's LR model is encoding realism and locality (ie., how it is making your, JesseM's, 1 and 2 explicit).

I also don't understand how realism and locality could be encoded (made clearly explicit) in any way other than the way Bell did it. In more or less recent threads this is what my fiddling with some way to specify the relationship between λ_a and λ_b without skewing the range of predictions had to do with.

So, unless GW is able to answer these questions, or demonstrate why a non-Bell-like LR formulation should be considered an LR formulation, then I'm compelled to take Bell's formulation as the archetypal LR model of entanglement -- thus, via Bell's demonstration, ruling out any and all LR models of entanglement.
[Note: The above considerations should make it clear why I think that Bell's theorem doesn't rule out the possibility that nature, including Bell tests and other realizations of quantum entanglement, is evolving locally. The fact that a viable model can be made which causally (via FTL or AAD) relates events in S1 and S2 is meaningless wrt what is actually happening at level(s) underlying instrumental results. Bottom line, we have no way of knowing, and assumptions/inferences of nonlocality aren't reasonable given the current situation in physics.]

 

[Gordon Watson]

Um, the whole point of the argument is that none of us believe you can come up with a local realist model that gives probabilities for P1-P8 which is "as one with QM".

There seems to be a major misunderstanding here. Not about my understanding of current beliefs, but about the next bit, which I'll rephrase as a question:

Q: Does GW really believe that he can come up with a local realist model that gives probabilities for P1-P8 which is "as one with QM"?

A: Yes; as has been shown in PDF2 (Table 1, Table 2, and notes thereto). So, in fact, GW believes that he has (beyond can) already come up with a local realist model that gives probabilities for P1-P8 which is "as one with QM"?

Let me explain: Table 1 is NOT based on QM; it is based solely on local realism of the L*R variety. So it must be interpreted in that context. Table 1 cannot be based on current understandings in QM, because the conditioning space (CS) includes three orientations (a, b, c).

[Aside (which is where my writing too often goes astray): Are you happy to have the space C in P(X|C) called the conditioning space; CS, for short?]

Further, any sum over Table 1 must be (initially, until the sum is reduced), a sum of of Ps with CSs abc. Thus SUM = P(X|abc) + P(Y|abc) + ... .

Since QM tests currently range over only two orientations, these summations cannot be QM results. They are L*R results. The QM results are delivered by reducing the CS to any two orientations.

PDF2: Table 2 gives every possible 2-orientation outcome. Appendix A has everyone worked out in full detail. All agree with QM.

To be clear here: The first Boundary Condition (B/C) on L*R is local realism. The second B/C is that all testable results must accord with QM. If one or both of these B/Cs is not met, the model fails.

PS:

QM can test over two orientations and one angle.

L*R can test over three orientations and two angles -- via a thought-experiment. For that is how L*R was developed.

I trust this might help to remove some of our "Table 1 and 2 and QM-outcome" differences?

Do you really not understand the really really basic point that all Bell inequalities are supposed to be claims about what must be true under local realism, not about what is predicted by the QM probabilities?

I do understand this point. I am wondering where I appeared to not understand?

The predicted QM probabilities are simply a B/C on the L*R predictions. If they are not "as one", then the model fails.

But look at your second point: "all Bell inequalities are supposed to be claims about what must be true under local realism." Again, where do we differ? They are supposed to be.

They are "supposed to be, and are widely believed to be" claims about what must be true under local realism.

Hence my bringing L*R here to see where it fails. FOR L*R is a local realistic model about what MUST be true about local realism.

And, just as in QM, L*R says Bell's inequalities cannot be constructed from within this local realistic world-view.

Tentative conclusion by GW (which is under test here at PF now): All Bell inequalities are supposed to be claims about what must be true under local realism. BUT L*R is definitely derived from local realism, local realism in its most general form, and Bell inequalities cannot be formulated therein.

Implied conclusions:

1. All Bell inequalities are supposed to be claims about what must be true under local realism, but they are not.

2. All Bell inequalities may (perhaps) be true under a naive form of local realism.

I will be follow-up on these matters, using the PDF2 material and expanded clarifying notes. I know there are many more questions that need to be answered.

 

[Gordon Watson]

the P

<snip>

regarding gordon watson's (gw) proposed lr model:

What i would like to see is a clear exposition and explanation of gw's lr ansatz. If, at that stage, it's ascertained to be nonrealistic, then whether it reproduces qm results is moot, because it wouldn't be an lr model.

That is, i still don't understand exactly how gw's lr model is encoding realism and locality (ie., how it is making your, jessem's, 1 and 2 explicit).

<snip>

[Note to Admins: If this expansion on the model is not permissible here, under PF guidelines, I'd be happy to lodge an application for it to go under Independent Research.]

ThomasT,

Does this help? I think it properly gets us to the nitty-gritty, and what I write might help others understand the simplicity beneath L*R. Or locate the defect, which is our goal!

1. In L*R there are no abstract entities; we work with real elements of physical reality, and every relevant element of such must appear in the relevant equations. Such elements include trajectories and angular momenta before any test. In the discussion here, we can proceed by considering test outcomes only. The WHY for our maneuvers needs some discussion of HVs, total-momentum orientations and perturbed trajectories in 3-space, projected onto 2-space.

2. In L*R there are just two types of spin-half particles in the world: Those that would yield + and those that would yield – when tested via an appropriate detector oriented a. Same for photons.

3. How come there are only two types? Well, in L*R, any test reveals an equivalence class (EC) to which that particle belongs.

4. So when we select a frame of reference (FoR), say a, we know that there are only two particle types that require consideration at that orientation: Those that will, or did, or could, yield +; those that will, or did, or could, yield – at this orientation.

5. Since the pristine-particle orientations are pairwise correlated, but otherwise random, the P(a+|a) from one set of twins to the next is 1/2. BUT, as a consequence of the pair-wise correlations of twins at their creation, if one twin (by test) belongs to a specific EC, the other twin belongs to a related EC, depending on the particlar correlation existing at their creation. In PDF2, the correlations are OPPOSITE. So if one particle EC-qualifies a+, its twin will qualify a–.

6. So here, if you are ready to maintain the discipline enforced by FoRs, you are now ready to derive Tables A1-A3 in PDF2. NB: Your readiness implies that you understand this fact: As you fill up the cells in your blank Table, you are repeatedly answering a question like this:

If this pristine particle would yield a + outcome if tested at a, what is the probability that its pristine twin would yield a + outcome if it were tested at b? And so on. Does this next comment help? Imagine the pristine twins to be stable correlated gyroscopes. We are going to perturb each, independently, via a measurement interaction. We are allowing that the perturbed 3-space orientation-based trajectories of both particles do this: They pass through a on their way to b; or vice versa; or they start between a and b and go their own correlated ways accordingly. In thought-testing many pairs, across many orientations, I know of no evidence that negates this trajectory-centric-view.

7. For the frame of reference selected (and you will be going through a, b, c in turn), the first number to write is 1/2; the Probability, for the outcome you are analyzing, occurring at that orientation. For the next orientation (say b), in this FoR-a Table (A1), you will write Cab if the sign is the same at that for the a frame; or Sab if the signs are different. For these are the related Ps that apply over the relevant trajectories. (We are just using Malus' Law, generalized.)

8. You will do the same for c in the a-FoR Table (A1).

9. Table A1 is now complete. So next do A2, then A3. Table 1 is the average over these 3 Tables. Table 2 (the crucial desideratum) follows, as per detailed equations in Appendix A, if you get lost.

10. Tip: The process is so straight-forward and robotic that I usually derive the results afresh each time that they're required. To get a flying start, copy the tables from PDF2 and blank out the answers only.

HOMEWORK: Since we are only dealing with squared cosines and sines, there is no excuse for not doing this as a way to begin answering the excellent point (imho) that you raise:

Time yourself over the above exercise, please. For this will be helpful info. I'm guessing that it takes maybe two hours max; especially with PDF2 to bail you out.

Please see if you differ from any result given in PDF2. This, of course, does not validate the model completely. There could be two wrongs making a supposed right.

But it will at least show the simplicity involved in L*R. You will have derived every possible QM outcome, in full accord with the QM-approved result. (The method, of course, not yet approved.) You will have followed the discipline which I believe FoRs bring to bear on the subject: Remembering that FoRs provide different accounts of the same phenomena.

So, seems to me like a fair investment for (maybe) two hours work.

I might even send you a PF subscription, if you find the defect:

Or you buy a 5-year one when you don't?

 

[Gordon Watson]

Here, things bug. Maybe this comes about because of a misunderstanding of the exact set-up, or about what exactly we are talking about, I don't know. There's no "double-valuedness" of any angle.

Consider the set-up as follows:

On Monday, Alice puts her analyser vertically and her detector clicks when she gets an "up" result. Bob puts his analyser at 45 degrees with the vertical (s times angle is then 22.5 degrees, right) towards the window of the room, and his detector clicks also when he gets an "up" result. An electronic circuit links Alice's and Bob's detector signals to a counter, which counts each time there is a simultaneous click on both detectors.
Carol starts the electron-pair source in the middle, and let's it generate 1 million electron-pairs during the afternoon.
Quantum mechanics predicts that at the end of the afternoon, the counter will read something like 73 000 counts.

On Tuesday, Alice leaves her installation in place, but Bob rotates his axis until it is horizontal (so s times angle is 45 degrees), with his "up" direction pointing towards the window.
When Carol starts the 1 million electron pair source again, quantum theory predicts that the counter will read 250 000 at the end of the afternoon.

On Wednesday, Alice, Bob and Carol go to a party.

On Thursday, Bob leaves his installation in the horizontal direction, but now Alice rotates her axis also in the direction of the window, over 45 degrees. Carol makes the source again run and produce 1 million electron pairs. Quantum mechanics predicts that the counter will read 73 000 counts.

The whole point is that the statistical mixture of the 1 million electron pairs is each time the same ; that the source didn't suffer any influence from the choice of settings.

So if on Friday, Alice and Bob randomly change their axes and we take data until we have 1 million events where Alice and Bob had aligned their axes as on Monday (so only considering those results when by coincidence Alice and Bob had their axes as on Monday), we expect to find statistically the same result as on Monday ; if we take data until we have 1 million events where Alice and Bob had aligned their axes as on Tuesday (so considering only those results where by coincidence Alice and Bob had their axes as on Tuesday), we expect to find the same result statistically as on Tuesday. And same for Thursday. Also, if by coincidence Alice and Bob put their axes parallel, we find that the counter reads 0.
The results will be the same if the source is generating statistically identical sets of events, independently of how the axes are set.Now, if we are to explain the results of Alice and Bob in a LR way, we have to assume that each pair sent out by the source must fall in 1 of 8 categories.

In the first category are the pairs which would give us a click in Alice's counter when it is vertical, and no click in Bob's counter when it is vertical ; that it would give us a click in Alice's counter when it was at 45 degrees, and no click in Bob's counter when it was at 45 degrees, and again that it would give a click in Alice's counter when at 90 degrees, and no click in Bob's counter when it was at 45 degrees. We write it as (+ + +). So events in this category will always give a click in Alice's counter and never one in Bob's counter.

and so on for the 7 other categories.

Note that there are no other possibilities: the 8 categories cover entirely the possibilities of the electron pair behaviour. It is for instance not possible that a pair wouldn't give a click in any Alice counter nor in any Bob counter. If a pair doesn't give a click in a vertical Alice counter, then it MUST give a click in a vertical Bob counter. So if we know the behaviour of a pair at Alice, we know that the behaviour at Bob's is complementary.

So the 1 million events must be subdivided in these 8 categories, with:

P1 * 1000000 = N1 the number of pairs in the first class,
P2 * 1000000 = N2 the number of pairs in the second class

etc...

Well, the number of pairs that belong to those that were counted on Monday are those in class 2 AND those in class 4. Each of the pairs in one of these classes will make the counter count, so we have that:

N2 + N4 = 73 000 up to statistical errors.

The counts on Tuesday are N3 + N4 = 250 000 up to statistical errors

The counts on Thursday are N3 + N7 = 73 000 up to statistical errors.

Well, you can't find such (positive) numbers N2, N3, N4, and N7.

Simply because if you add the counts on Monday and those on Thursday,

N2 + N3 + N4 + N7 = 146 000

and the counts on Tuesday are only N3 + N4 and they are LARGER: 250 000.

There's no "double angledness" or whatever here. There are specific measurements, with specific outcomes, and you CAN'T explain them with a pre-determined mixture of events. That's the point.

Excuse delayed reply here. I answered a related post earlier, but wanted to be sure that the issues here were clearly covered.

I didn't understand the reference to windows, and some settings pointing to them. For example, this confused me: "with his "up" direction pointing towards the window."

However, I believe that the outcomes relate to QM outcomes, as given in that other reply. So, per Table 2 in PDF2, we are not disagreeing about valid QM results.

However: With the numbers that were meant to be "pedagogical" -- they are numbers derived from L*R. Such numbers do not deliver the QM numbers directly. Instead they deliver the numbers that correctly relate to the "L*R 3-orientations, 2-angles, 1 bi-angle" thought-experiments that characterize L*R.

To get the QM numbers that you seek to check, the "L*R 3-orientations, 2-angles, 1 bi-angle" results must be reduced to a result that QM relates to. QM results involve and relate to "2-orientations, 1-angle" tests.

These reductions are fully detailed in Appendix A of PDF2. They yield (as shown in Table 2 of PDF2), correctly, every possible QM number that relates to the experiment.

On this basis, I'd be pleased if you would reconsider the "pedagogical" merits of my P1-P8 L*R-based numbers.

Many thanks.

 

[ThomasT]

Thanks GW.

Ok, we're only interested in predicting coincidental photon flux (++), so your LR formula for the expectation value in, say, the Aspect experimental setting reduces, in conventional notation, to,

P(A,B) = (cos²θ)/2,

Is that correct?

If so, then in this form it doesn't qualify as an LR model. The various 'Cab', etc. in your Tables also don't qualify as LR.

I mean, it looks as if you've just added λ (your s) to the qm formula to define 'Cab' etc., and then just omitted it (presumably because λ is continuous from a to b, and a randomly occurring value) to calculate the values for various θ.
If so, then your model isn't an LR model.

So if you could take us back to some point in your derivation where your formulation encodes realism and locality (that is, where λ, your s, and a locality condition actually have something to do with the calculations), then that would be helpful in evaluating your claim.

Without that, it doesn't matter whether or not your tables are able to reproduce qm results.

 

[Gordon Watson]

Well this thread is the result of a request by JenniT that he/she COULD generate 8 numbers P1...P8 such that it corresponded to the quantum predictions. This is clearly impossible, but up to now JenniT has been claiming otherwise.

As far as I can see, JenniT has offered only ONE proposal involving the "8 numbers" referred to above.

Those numbers are as given in Table 1 of PDF2; as general functions of C and S.

I believe the alleged impossibility derives from a serious misunderstanding.

That misunderstanding is this: The 8 numbers in PDF2, the 8 numbers provided by JenniT, are numbers within L*R. See Table 1 of PDF2.

They are not QM numbers. The numbers that relate to QM are derived from the L*R numbers. These QM numbers are given in Table 2; there are more than 8 of them, and all agree with QM.

I further address this issue below.

His/her first attempt gave:

QM: 0.25, 0.073, 0.073 (for spin-1/2 particles and axes 0 degrees, 45 degrees and 90 degrees) and JenniT produced a first set of 8 numbers such that the numbers that came out were 0.125, 0.073 and 0.073, and there was a lot of hot air about a claim that these WERE the right results because of "an average that had to be taken over two different angles" without ever having cleared this up.

Here the above-mentioned misunderstanding is explicit. The numbers 0.25, 0.073, 0.073 are the correct QM numbers. They are equally derived from QM and L*R.

The numbers 0.125, 0.073 and 0.073 are L*R numbers. Two (0.073) are equally QM numbers. The 0.125 number is specifically L*R. Let us see what it relates to; then if it is correct:

The L*R model states that, in the calculation delivering 0.125, 0.073 and 0.073, the 0.125 is an average over two values. In the given example, the two values are 0 and 90. (And to be noted in passing, the experiment was carried out on the 90 setting, and L*R gave the correct result: 0.25.)

But is 0.125 the average of an 0 and a 90 setting; calling them ab-1 and ab-2 respectively?

Average = [S(ab-1)/2 + S(ab-2)/2]/2 = [0 + 0.25]/2 = 0.125.

I trust this goes some way to clearing up the "two angles" that L*R deals with correctly. For we have this FACT: L*R gives numbers BEYOND QM, plus every possible QM number. L*R said that this would be the case from day one.

In that the QM numbers are correctly delivered, it seems to me that the real question remains: Is L*R truly local and realistic? This seems to me to be the question that ThomasT seeks to address.

Now we seem to have ANOTHER proposal by JenniT where he/she claims this time to HAVE produced 8 numbers such that the predictions come out to be:

0.25, 0.073 and 0.073

after some algebra.

As this is algebraically impossible, we ask him to give us the 8 numerical values, and show how they comply with the above calculation.

As stated above: As far as I can see, JenniT has offered only ONE proposal involving the "8 numbers" referred to above.

Those numbers are as given in Table 1 of PDF2; as general functions of C and S.

Further: The requisite algebra, to derive the 3 numbers above, is spelled out in Appendix A of PDF2, and summarized in Table 2.

As I interpret your example with the window, let us take: ab = 90, ac = bc = 45.

From PDF2, from the given worked example but inserting specific numbers:

Pab(++|ab) = Sab/2 = 0.25.

In similar manner, we also have:

P(ac++|ac) = Sac/2 = 0.0732

P(bc++|bc) = Sbc/2 = 0.0732

These are the correct QM and L*R predictions!

I see no algebraic impossibilities here.

You're right, but we're dealing with somebody who claims he knows how to make one.

Somebody who is happily and openly checking and learning within the PF community, to find possible errors. Someone who is very appreciative of your contributions, and many others.

Thus far, with many questions yet to be answered, I find unfortunate misunderstandings (about the L*R model, which is thus far unchanged), and I accept my role in many such. But unfortunate misunderstandings do not constitute errors; though errors there may be.

 

[Gordon Watson]

Thanks GW.


Ok, we're only interested in predicting coincidental photon flux (++), so your LR formula for the expectation value in, say, the Aspect experimental setting reduces, in conventional notation, to,

P(A,B) = (cos²θ)/2,

Is that correct?

I'm confused. The L*R example deals with an example in Zakurai, originally introduced by vanesch, with spin-half particles.

OK, Aspect deals with photons; s = 1 for photons.

And as you can see, L*R deals with both. But I suggest we stick with that s = 1/2 example for now.

If so, then in this form it doesn't qualify as an LR model. The various 'Cab', etc. in your Tables also don't qualify as LR.

Could you carefully elaborate this please. You may be on the right track, but I don't yet see it from these brief remarks. Thanks.

I mean, it looks as if you've just added λ (your s) to the qm formula to define 'Cab' etc., and then just omitted it (presumably because λ is continuous from a to b, and a randomly occurring value) to calculate the values for various θ.
If so, then your model isn't an LR model.

You write: λ (your s) ?

My s = intrinsic spin, as defined from day one.

Could you carefully elaborate this please. You may be on the right track, but I don't yet see it from these brief remarks. Thanks.

So if you could take us back to some point in your derivation where your formulation encodes realism and locality (that is, where λ, your s, and a locality condition actually have something to do with the calculations), then that would be helpful in evaluating your claim.

There's wrong bits here; they may be clouding your valid point of view; and my view of it!

You write: "where λ, your s," ?

My s = intrinsic spin, as defined from day one.

Without that, it doesn't matter whether or not your tables are able to reproduce qm results.

Let us see; after you've clarified and expanded your text. OK? And thanks.

 

[ThomasT]

I'm confused. The L*R example deals with an example in Zakurai, originally introduced by vanesch, with spin-half particles.

OK, Aspect deals with photons; s = 1 for photons.

And as you can see, L*R deals with both. But I suggest we stick with that s = 1/2 example for now.

Ok, Bell's LR formula for the singlet state expectation value is,

P(a,b) = ∫dλρ(λ)A(a,λ)B(b,λ),

and the qm formula is,

< σ₁ ∙ a σ₂ ∙ b > = - a ∙ b = - cosθ, where θ is equivalent to your ab.


What is your LR formula for the singlet state expectation value?

Could you carefully elaborate this please. You may be on the right track, but I don't yet see it from these brief remarks. Thanks.

According to your paper, Cab; etc. = cos²sab; etc. Sab; etc. = sin²sab; etc. ab = angle between orientations a and b; etc. s = intrinsic particle spin.

So, how is Cab, Sab (etc.) to be evaluated?

You write: λ (your s) ?

My s = intrinsic spin, as defined from day one.

Could you carefully elaborate this please. You may be on the right track, but I don't yet see it from these brief remarks. Thanks.

λ is the conventional notation for the hidden variable. Isn't s your hidden variable? Is it affecting the value of Cab? How? If not, then I don't understand what s is doing in Cab.