Thoughts about Bell, Bohm, Christian, et al.

[minkwe]

Outcomes exist in reality, but are not defined by QM.

I wasn't asking you what QM defines, but whether individual outcomes are understood to exist in QM. Of course QM does not predict individual outcomes but that does not mean the outcomes do not exist, nor does it mean QM doesn't have symbols for those outcomes.

I would not write them down mathematically, because I see no reason for this. But, ok, I could write . But in such a case I would prefer the verbal description.

Of course you would prefer that, because it is a rhetorical device which affords the convenience of being vague. But let me Press you some more there, for a single particle heading toward a station oriented along vector "a", what exactly do your terms A, B, AB, mean? In other words,



Is too imprecise. It does not identify the outcomes with the magnet directions. I seek two expressions:

1) should represent the fact, stated in the definition of the problem, that the outcomes are ±1 for a single particle measured along "a"
2) the second should represent the fact that, for two such entangled spin-1/2 particles, measured along the same axis "a", the outcomes are opposite.

Do you have any problem with these:?



That is, do you disagree that these expressions are universally true, even for QM.

Those two facts are the premises of the paper, and are universally true even for QM, and involve no ambiguity that should require a treatment using probabilities. Can you agree to that?

I do not think that the expression (1) somehow by symbol magic enforces locality. I think that the context - the EPR argument used to describe what is the λ - is what requires that (1) is local.

Do you understand the difference between:

* If λ is local, then A(a,λ) = +/-1
* If A(a,λ) = +/-1, then λ is local
If so, then you should also understand exactly why it is important to distinguish between necessary and sufficient conditions in mathematical arguments. It ultimately determines how logically sound our arguments are.

No, I claim that QM does not have any formulas to compute the outcomes.

I did not ask you for an expression to compute the outcomes. I asked you for an expression representing mathematically the basic facts of the experiment as described by Bell himself, in a manner which applies just as well to QM as to local theories.

[minkwe]

Why awaiting your response to my previous point, I would like to come back to this point I made earlier.

Do you agree that equation (1) would be exactly the same even if λ represented non-local hidden variables? Even if we all agree that when Bell wrote equation (1) he was thinking about local hidden variables. But my question is not about what Bell was thinking. Would equation (1) be different if lambda represented non-local hidden variables?

Second point I want to raise is that if you agree with my previous answer to my own question, that we could represent the facts of the experiment described in the second paragraph on page 1 with
,
where represents the outcome of a single particle at Alice's station, when measured along direction "a", and represents the outcome of a single particle at Alice's station, when measured along direction "b".
then we could also introduce λ in this expression to give
,
And hopefully, you don't have any problem with this expression either. Now obviously Bell was thinking about λ as being local hidden variables, but hopefully you also agree that the expression will be exactly the same if λ represented non-local hidden variables. If all of this is acceptable to you, we can proceed to equation (2).

[AnotherGuest]

if we look ahead to (13), we see that Bell himself deduces that A(a, lambda) = - B(b, lambda) from the facts that P(a, a) = -1, A and B take values in {-1, +1}, and rho is a probability density.

In equation (1), lambda represents all those hidden variables, which, together with the settings a and b, determine the measurement outcomes. Bell assumes them to have a probability distribution which doesn't depend on a and b. You could think of them as being located wherever you like. They could be a whole list of hidden variables, living in different places. Eg lambda is a vector wth three components: hidden variables in the three locations in the story.

[Schmelzer]

But let me Press you some more there, for a single particle heading toward a station oriented along vector "a", what exactly do your terms A, B, AB, mean?

For a single particle A would mean the measurement result, which is in , and B together with AB would not have a meaning, because this would presuppose the other particle.

Is too imprecise. It does not identify the outcomes with the magnet directions. I seek two expressions:
1) should represent the fact, stated in the definition of the problem, that the outcomes are ±1 for a single particle measured along "a"
2) the second should represent the fact that, for two such entangled spin-1/2 particles, measured along the same axis "a", the outcomes are opposite.

I cannot follow. (1) is clearly represented by . And (2) by

Do you have any problem with these:?


That is, do you disagree that these expressions are universally true, even for QM.

I actually see no problem for QM. I have a general philosophical problem with statements being "universally true", independent of a particular theory.

Those two facts are the premises of the paper, and are universally true even for QM, and involve no ambiguity that should require a treatment using probabilities. Can you agree to that?

No. The connection between A and a is, of course ambigious and requires a treatment using probabilities. suggests some functional dependence for which there is no base in QM.

Do you understand the difference between:
* If λ is local, then A(a,λ) = +/-1
* If A(a,λ) = +/-1, then λ is local
If so, then you should also understand exactly why it is important to distinguish between necessary and sufficient conditions in mathematical arguments. It ultimately determines how logically sound our arguments are.

Oh, sorry, I'm completely stupid, never heard about such a thing like a difference between necessary and sufficient conditions, can you explain? Or what do you want to hear?

I asked you for an expression representing mathematically the basic facts of the experiment as described by Bell himself, in a manner which applies just as well to QM as to local theories.

And why do you think I should find such an animal?

Do you agree that equation (1) would be exactly the same even if λ represented non-local hidden variables? Even if we all agree that when Bell wrote equation (1) he was thinking about local hidden variables.

The question is incorrect, because "local hidden variables" as well as "non-local hidden variables" is simply sloppy language. If I have accepted that sloppy language in earlier answers, I should remove this as inaccurate. What is local or nonlocal is the realistic theory.

Locality is used (and necessary) for the EPR argument, and gives two results:
1.) The values A, B are predefined, thus, there exists some hidden variables so that A=A(a,b,λ) and B=B(a,b,λ).
2.) These λ have a probability distribution which cannot depend on a,b, because λ is predefined, that means, defined before a, b come into existence.
3.) A should not depend on b and B not on a.

Second point I want to raise is that if you agree with my previous answer to my own question, that we could represent the facts of the experiment described in the second paragraph on page 1 with
,
where represents the outcome of a single particle at Alice's station, when measured along direction "a", and represents the outcome of a single particle at Alice's station, when measured along direction "b".
then we could also introduce λ in this expression to give
,

If you introduce it, you would have to describe what they mean. And, whatever you use at this description, it is highly probable that, then, the two equations mean something very different than the equations in Bell's paper, where the λ is defined using the EPR argument, which leads to some properties of λ which are important.

Note: Mathematics is only language, a little bit more formalized than everyday language, but this does not change the fact that formulas have a meaning only in a context.

And hopefully, you don't have any problem with this expression either. Now obviously Bell was thinking about λ as being local hidden variables, but hopefully you also agree that the expression will be exactly the same if λ represented non-local hidden variables. If all of this is acceptable to you, we can proceed to equation (2).

I have the problem that you have completely changed the context, thus, the object λ derived from the EPR argument (which has properties which follow from the derivation) is replaced by some abstract λ without any properties. That means, the context is modified, and therefore the formula obtains a different meaning. If you interpret, for example, λ as "everything which is somehow responsible for the output A, B", and don't have the context of the EPR argument, you could, for example, now write formula (2) with or some other manipulation.

[minkwe]

No. The connection between A and a is, of course ambigious and requires a treatment using probabilities. suggests some functional dependence for which there is no base in QM.

If you refuse to agree to the basic facts:
,
where represents the outcome of a single particle at Alice's station, when measured along direction "a", and represents the outcome of a single particle at Alice's station, when measured along direction "b", and you refuse to admit the self evident fact that , is an accurate (and relevant, and in context), representation of Bell's statenents on page 1, then we are done here. I don't think you have the capacity to understand Bell's theorem. You think you know what my next argument is going to be, so you try hard to pre-empt and block it not realizing that your current arguments are not making any sense. I haven't said anything about equation 2 yet. You are more interested in vague argumentation than clear and precise presentation and explication of the issues involved.


I don't have time to waste, so feel free to continue the discussion with others. I'm done. Good luck in your endeavours.

[AnotherGuest]

I am puzzled about the exchange so far between minkwe and Schmelzer.

(1) Once we have written down some mathematics assumptions we can start doing mathematics. From the informally made mathematical assumptions discussed so far, it seems to me clear that for all , we have that . So and are actually the same functions. And Bell says as much, later in the paper.

(2) If the mathematical assumptions made so far would also apply to some non-local hidden variable theories, then the mathematical conclusions would apply to those theories, too. So the mathematical theorem (which I guess is the famous Bell inequality (15) in http://www.drchinese.com/David/Bell_Compact.pdf) is actually more general than most people think.

I don't see why Schmelzer has a problem with either of these observations.

[Schmelzer]

No. The connection between A and a is, of course ambigious and requires a treatment using probabilities. suggests some functional dependence for which there is no base in QM.

If you refuse to agree to the basic facts:
,
where represents the outcome of a single particle at Alice's station, when measured along direction "a", and represents the outcome of a single particle at Alice's station, when measured along direction "b",

I don't refuse to agree with this description. My point was that, if we use this description, we do not have this formula alone, but this formula in this particular context: represents the outcome of a single particle at Alice's station, when measured along direction "a". Taken out of this context, the formula becomes open to misinterpretations.

and you refuse to admit the self evident fact that , is an accurate (and relevant, and in context), representation of Bell's statenents on page 1, then we are done here.

Again, I do not refuse to admit this fact , all I do is to insist that this formula should not be taken out of the context. And this context is given by Bell's statements on p.1-2 of his paper, by the derivation of the existence of lambda based on the EPR argument.

I don't think you have the capacity to understand Bell's theorem.

As usual, no arguments left -> personal insults.

You think you know what my next argument is going to be

And your refusal to surprise me by presenting a completely different, new argument seems to indicate that this guess was not that wrong.

So, for the readers I try to summarize the central point.

First, Bell uses , as a basic fact about single experimental results predicted by QM. Then, he applies the EPR argument: Given that we can predict by measuring , which, because of locality, does not influence at all. Thus the EPR criterion of reality is applicable, thus, is not a random number possibly generated randomly at measurement time, but predefined.

Predefined means defined before the experiment was done, before a was chosen. And now the mathematical formalism is introduced to describe what this "predefined" means. It means that for every preparation where exists and some functions which define the outcomes of the measurement for all possible values a,b. The preparation procedure creates somehow this with some probability distribution which, because is predefined, that means defined before a,b have been chosen, does not depend on a,b too.

My guess was that minkwe has tried to get rid of this context, and put the expressions into a different context, where is no longer the set of parameters which predefines A and B, but is simply some general set of parameters which defines A and B at the moment of the measurement. With such a different context, there would be nothing to object against a measure which depends on a and b. But in this case, Bell's theorem would be invalid.

[minkwe]

My guess was that minkwe has tried to get rid of this context, and put the expressions into a different context, where is no longer the set of parameters which predefines A and B, but is simply some general set of parameters which defines A and B at the moment of the measurement. With such a different context, there would be nothing to object against a measure which depends on a and b. But in this case, Bell's theorem would be invalid.

Not only was your guess completely wrong, you are all over the place, objecting and then accepting the same things you've been objecting about, wasting my time. I don't think it is possible to have an intelligent insightful discussion with you, so unfortunately for you, you won't get to see what my argument is. I'm done. Feel free to wallow in your glee.

For anyone else following, there's a question Schmelzer has failed to answer, and there is a reason why. When I asked earlier for an expression that represents the facts stated by Bell in the third sentence of page (1), he writes , which reveals a lot, as it is a superfluous introduction of probabilities into a situation that does not warrant one. He should have just written , but that would have meant conceding the fact that applies to QM just as well. Worse, he proceeds to make the incredible claim that:

The connection between A and a is, of course ambigious and requires a treatment using probabilities. suggests some functional dependence for which there is no base in QM.

Seriously?! is a symbol representing the outcome at Alice's station with her magnet oriented along axis "a". , means that her outcome is either +1, or -1. According to QM, the outcomes are opposite for the same axis "a", thus , so the ambiguity Schmelzer is dreaming about is completely irrelevant and inexistent.

According to Bell's description on page 1 and up to equation (1), For a single particle heading towards a SG magnet oriented along vector "a", where λ is a local hidden variable, the outcome can be represented as A(a,λ) = ±1, according to Bell himself. The unanswered question is
"How would this expression be different, if λ was a non-local hidden variable rather than a local hidden variable?"
And for two spin-half entangled particles heading in opposite directions and measured along the same axis "a", Bell's description is equivalent to A(a,λ) = -B(a,λ) = ±1.
These expressions are exactly equivalent to . Still no answer to the question "How would this expression be different, if λ was a non-local hidden variable rather than a local hidden variable?" this expression is not more or less open to misrepresentation than Bell's, and despite claiming the expression was open to misrepresentation, Ilja has not shown a single misrepresentation of Bell in anything I've posted, nothing other than imagined misrepresentations,and false guesses about what I might be planning to do.

He thinks "the question is incorrect, because "local hidden variables" as well as "non-local hidden variables" is simply sloppy language." Really?! If we shouldn't talk about local hidden variables, or non-local hidden variables, what language is accpetable to Schmelzer. Despite claims about "out-of-context" etc, the question is relevant and in context, as far as Schmelzer wants to suggest that equation (1) of Bell's paper somehow incorporates the concept of locality.

Ultimately, the fact remains:
Bell's equation (1) would not be any different if λ was a non-local hidden variable therefore it is fantasy to think that equation (1) contains any concept of locality. If as is now obvious, non-local hidden variables λ would also give us equation (1), then any negation of equation (1) can not be used reject only "local hidden variables", it would have to apply just as well to "non-local hidden variables."

[Joy Christian]

Bell's equation (1) would not be any different if λ was a non-local hidden variable therefore it is fantasy to think that equation one contains any concept of locality.

Correct. However, to be fair to Bell, he did not think that his equation (1) by itself contained the concept of locality. He therefore defines the concept of locality separately, and quite precisely, as follows (and I paraphrase): The measurement outcomes A(a, λ ) = +/-1 and B(b, λ) = +/-1 are said to be locally explicable if A(a, λ) does not depend on either b or B, and likewise B(b, λ) does not depend on either a or A. This of course puts severe restrictions on what the "functions" λ can be.

To any rational and intelligent reader it should have been obvious what you were asking, at least after you clarified your statement in response to my first question.

[minkwe]

Correct. However, to be fair to Bell, he did not think that his equation (1) by itself contained the concept of locality.

We are going one equation at a time. I never claimed that Bell intended equation (1) to represent locality, I simply affirmatively stated that Equation (1) will not be different if λ were non-local.

He therefore defines the concept of locality separately, and quite precisely, as follows (and I paraphrase): The measurement outcomes A(a, λ ) = +/-1 and B(b, λ) = +/-1 are said to be locally explicable if A(a, λ) does not depend on either b or B, and likewise B(b, λ) does not depend on either a or A. This of course puts severe restrictions on what the "functions" λ can be.

Sure, but even that prescription does not prevent λ from being non-local in a general sense. It might prohibit dependence on remote settings, but it does not prohibit dependence on any other remote information present at a location outside the light code of A. And if you consider just a single particle being measured along a single axis, that definition of "locality" is incoherent, as there is no "b" axis, or an infinite number of "b", "c", "d" ... axis that could be chosen at an infinite number of different space-time coordinates. You could instead group them all to a single environment vector "E" to say locality means A(a,λ) should not depend on E, ie A(a, λ, E), where E a vector representing all environmental properties outside the light cone of Alice which could effectively be summarized into A(a, γ), where γ = (λ, E) represents everything else which together with "a" is responsible for the outcome. We still arrive at the same expressions as equation (1), albeit with an obviously non-local variable γ. And the rest of the derivation can proceed just the same.

So either we need the non-local setting to define locality in a manner which is incoherent for a single particle, or we define locality coherently for a single particle in a manner that is also consistent for any number of particles, and end up with the same equations.

[Heinera]

Still no answer to the question "How would this expression be different, if λ was a non-local hidden variable rather than a local hidden variable?"

And what is your mathematical definition of a non-local hidden variable?

[minkwe]

Still no answer to the question "How would this expression be different, if λ was a non-local hidden variable rather than a local hidden variable?"

And what is your mathematical definition of a non-local hidden variable?

Wasn't that clearly described in my previous post:

Sure, but even that prescription does not prevent λ from being non-local in a general sense. It might prohibit dependence on remote settings, but it does not prohibit dependence on any other remote information present at a location outside the light code of A. And if you consider just a single particle being measured along a single axis, that definition of "locality" is incoherent, as there is no "b" axis, or an infinite number of "b", "c", "d" ... axis that could be chosen at an infinite number of different space-time coordinates. You could instead group them all to a single environment vector "E" to say locality means A(a,λ) should not depend on E, ie A(a, λ, E), where E a vector representing all environmental properties outside the light cone of Alice which could effectively be summarized into A(a, γ), where γ = (λ, E) represents everything else which together with "a" is responsible for the outcome. We still arrive at the same expressions as equation (1), albeit with an obviously non-local variable γ. And the rest of the derivation can proceed just the same.

γ is non-local hidden variable if it represents information outside of Alice's light cone. A theory which makes use of such variables to compute outcomes will be a non-local hidden variable theory. If Alice's outcome depends on γ, then A(a, γ) = -B(a, γ) = ±1 applies to such non-local theories just the same. This definition is consistent for both single particles and multiples of particles.

[Heinera]

I aksed for a mathematical definition, i.e., how would you formalize this with mathematical expressions?

[minkwe]

I aksed for a mathematical definition, i.e., how would you formalize this with mathematical expressions?

Formalize what exactly, that is not already clearly formalized in my previous response?

γ is non-local hidden variable if it represents information outside of Alice's light cone. A theory which defines Alices outcome as dependent on γ, is non-local. What more do you want.

Do you also believe Bell's equation (1) will be different if if λ was a non-local hidden variable rather than a local one? I say it would be exactly the same. If you disagree, then maybe you should be the one to state what the expression would be for a non-local hidden variable γ, where γ represents all information outside of Alice's light-cone.

[Heinera]

E.g., do you mean that the hidden vairable is a function ? In that case Bell's integral equations don't hold, so his proof is not valid for that case.

[minkwe]

E.g., do you mean that the hidden vairable is a function ? In that case Bell's integral equations don't hold, so his proof is not valid for that case.

By non-local I mean any information outside the light cone of Alice. As I've explained in my previous posts, defining locality/non-local based on specific settings at a specific location is incoherent, as you can't then apply it to a single particle. But my definition is more consistent as it applies to any number of particles. So again:

Let γ represent information outside Alice's light cone. γ is non-local to Alice. If a theory defines the outcome of her measurement along axis "a", as dependent on γ, eg , then that theory is non-local. Conversely, if λ represents only information present within Alice's light-cone, then λ is a local hidden variable, and any theory which defines the outcomes as dependent only on the local information λ, eg , is a local hidden variable theory. Equation (1) of Bell's paper, has no distinguishing feature that should be different for local/non-local hidden variable theory, other than what we might have in mind about the meaning of the symbols. The form of the expressions is exactly the same for local as for non-local theories. In other words, any suggestion that equation (1) somehow embodies a locality assumption is false. This is not controversial at all, we're not yet talking about the integral of equation (2).

[Heinera]

[ Equation (1) of Bell's paper, has no distinguishing feature that shoulIn other words, any suggestion that equation (1) somehow embodies a locality assumption is false. This is not controversial at all, we're not yet talking about the integral of equation (2).

The whole premise of his paper is based on a locality assumption, as is made very clear in the very next sentence following eq. (1). On the other hand, if the hidden variable would be a function of a and b, eq. (2) no longer holds. Hence, there is no longer a proof, so Bell's theorem does not apply to that case.

[Schmelzer]

... so unfortunately for you, you won't get to see what my argument is.

YMMD.

He should have just written , but that would have meant conceding the fact that applies to QM just as well. Worse, he proceeds to make the incredible claim that:

The connection between A and a is, of course ambigious and requires a treatment using probabilities. suggests some functional dependence for which there is no base in QM.

Seriously?!

Of course. If you repeat exactly the same experiment another time, you will probably obtain a different result. QM tells us about the statistics. without any clarification that only a single experimental outcome is considered would suggest something different, namely that a repetition of the same experiment would give the same result .

is a symbol representing the outcome at Alice's station with her magnet oriented along axis "a". , means that her outcome is either +1, or -1. According to QM, the outcomes are opposite for the same axis "a", thus , so the ambiguity Schmelzer is dreaming about is completely irrelevant and inexistent.

If one adds a sufficient long text to clarify the context, and, therefore, the meaning, the ambiguity can be removed. No problem. It does not mean that it does not exist without such a specification. taken alone does not clarify enough to tell if it is an unproblematic description about a single experiment or a false description about what QM tells us about the experiment, namely that there exists a function which describes the outcomes of such experiments if the measurement is done in direction a.

The unanswered question is "How would this expression be different, if λ was a non-local hidden variable rather than a local hidden variable?"
He thinks "the question is incorrect, because "local hidden variables" as well as "non-local hidden variables" is simply sloppy language." Really?! If we shouldn't talk about local hidden variables, or non-local hidden variables, what language is accpetable to Schmelzer.

Talk about well-defined notions used in the Bell text.

Variables can be localized - a, b, A, B are localized, have well-defined positions in space and time in this experiment. What is local are causal influences, or theories if the allow only local causal influences. But, ok, let's allow some sloppy language here, and assum that what I have named here "localized" could be named as well "local". Fine. But in this case nothing suggests that λ is local. The only thing what is known about λ is that it is predefined, it exists already before a and b have been chosen. Even if it would be localized, this is one half of the whole spacetime, not very much localized. λ can be as well some element of a space of functions on configuration space or phase space. Then it would not have a localization in space at all. How could this function be named "local" would be completely unclear.

Ultimately, the fact remains: Bell's equation (1) would not be any different if λ was a non-local hidden variable therefore it is fantasy to think that equation (1) contains any concept of locality. If as is now obvious, non-local hidden variables λ would also give us equation (1), then any negation of equation (1) can not be used reject only "local hidden variables", it would have to apply just as well to "non-local hidden variables."

Ultimately, the fact remains: Bell's equation (1) would not be any different if λ was a dishonest hidden variable therefore it is fantasy to think that equation (1) contains any concept of honesty. Define what means that λ is a non-local variable resp. a local one. Or, even better, don't mingle the notions "(non-)local hidden variable theory" with "(non-)local hidden variables". (Of course, not to a subhuman like me, I'm not dignified to receive such a propagation, but may be for the Reader? )

[AnotherGuest]

You guys are quarelling about the meaning of the word "local". Could you leave that aside for the moment, and proceed? Bell has explained *what* he has assumed. His assumptions include everything that most people are happy to call a "local hidden variable theory". Minkwe feels that the assumptions also cover some non-local theories. But he hasn't explained yet why that should later spoil "Bell's theorem".

If Bell's theorem is "this class of models cannot reproduce the singlet correlations", and if "this class of models" does include all genuine LHV models, then it follows (if Bell's theorem is true) that "no LHV model can reproduce the singlet correlations".

By the way, so far we might just as well consider the co-domain of A and B = -A as being {-1, +1} as a subset of the real line, the complex numbers, the quaternions, the octonions ... or of some other fancy space.

[minkwe]

... so unfortunately for you, you won't get to see what my argument is.

YMMD.

Sorry Ilya, I'm no longer interested in anything you have to say, so I'm adding you to my ignore list.