What do violations of Bell's inequalities tell us about nature?

[ttn]

As far as I'm concerned, his definition of locality requires the existence of the probabilities of the form p(a,b,\lambda), so if they don't exist (which definitely is the case in QM even for the simple case of a free 1D particle), then the definition can't be applied. Up to now, p(a,b,\psi) is only a purely formal expression void of any precise meaning. In particular, it's not a probability.

I'm sorry, but... what the heck are you talking about? Are you really saying that ordinary QM doesn't allow you to calculate what the probabilities of various possible measurement outcomes are, in terms of the state ψ of the system in question? That's the one thing that orthodox QM is unquestionably, uncontroversially good for!

Maybe the issue has to do with what I assume(d) was just a typo? Namely: it's not p(A,B,\lambda) but rather p(A,B|\lambda) -- or, as I indicated before, slightly more precisely, p_{\lambda}(A,B).

By the way: I was looking for that paper you suggested, but i don't find it on the internet. (Apart from that, i don't know french, so i probably couldn't read it?) Can you point me to a source? I have access to most journals.

You mean "la nouvelle cuisine"? First off, it's not in French. Only the title. =) The easiest place to find it is in the 2nd edition of "Speakable and Unspeakable in QM", the book collection of Bell's papers on the foundations of QM. The book is on google books, but unfortunately this particular paper isn't included. And I also couldn't find the paper online. If you don't have access to a library that has the actual book (though the book is cheap and brilliant so maybe it's a good excuse to spring for a copy), my paper quotes a lot from it and will certainly allow you to understand Bell's definition:

http://arxiv.org/abs/0707.0401

 

[stevendaryl]

Not in the definition of locality! (Yes, such a thing does come up in the derivation of the inequality, though.)

I don't think he actually gave a definition of "locality". The way I interpreted what he was doing was describing a class of models, and then proving that no model in that class could reproduce the predictions of quantum mechanics. If he gave an explicit definition of what "local" means, I didn't see one.

I am highly skeptical of this. First of all, the claim that was made here was that Bell's definition of locality is inapplicable if the space of λs is unmeasureable.

Maybe it would help the discussion if you wrote down what you consider Bell's definition of "local". What I have seen is this:

• Assume in an EPR-type experiment (assume the spin-1/2 version for definiteness) involving Alice and Bob that there is a deterministic function A(\hat{a}, \hat{b}, \lambda) giving Alice's result (+1 or -1) as a function of Alice's choice of detector orientation, \hat{a}, Bob's choice of detector orientation, \hat{b}, and some unknown parameter \lambda shared by the two particles by virtue of their having been produced as a twin-pair. Similarly, assume a deterministic function B(\hat{a}, \hat{b}, \lambda) giving Bob's result.
• Then, in terms of such a model, we can call the model "local", if A(\hat{a}, \hat{b}, \lambda) does not depend on \hat{b}, and B(\hat{a}, \hat{b}, \lambda) does not depend on \hat{a}. In other words, Alice's result is A(\hat{a}, \lambda) and Bob's result is B(\hat{b}, \lambda).
• Theorem, there are no such functions A(\hat{a}, \lambda) and B(\hat{b}, \lambda).

The proof of the theorem assumes that the unknown hidden variable \lambda is measurable; in particular, that it makes sense to talk about things such as "the probability that \lambda lies in some range such that A(\hat{a},\lambda) = B(\hat{a},\lambda)" for various choices of \hat{a} and \hat{b}. Pitowky showed that if you don't assume measurability of \lambda, then the EPR correlations can be explained in terms of a non-measurable function F(\hat{r}) where \hat{r} is a unit vector (or alternatively, a point on the unit sphere), with the properties that:
(This is from memory, so I might be screwing these up):

• F(\hat{r}) is always either +1 or -1.
• \langle F \rangle = \frac{1}{2}: The expectation value, over all possible values of \hat{r}, of F(\hat{r}) is 0.
• If \hat{r_1} is held fixed, and \hat{r_2} is randomly chosen so that the angle between \hat{r_1} and \hat{r_2} is \theta, then the probability that F(\hat{r_1}) = F(\hat{r_2}) is cos^2(\dfrac{\theta}{2})

Mathematically, you can prove that such functions exist (with the notion of "probability" in the above being flat lebesque measure on the set of possibilities). Pitowksy called it a "spin-1/2 function".But it's not a very natural function, and is not likely to be physically relevant.

But anyway, were it true, then wouldn't it follow that it was impossible to meaningfully assert that Pitowsky's model is local?

It's explicitly local: When a twin pair is created, a hidden variable, F is generated. Then when Alice later measures the spin along axis \hat{a}, she deterministically gets the result F(\hat{a}). When Bob measures the spin of the other particle, he deterministically gets -F(\hat{b})

I'd even be willing to bet real money that this isn't right -- that is, that there's no genuine example of a local theory sharing QM's predictions here. If it were true, it would indeed be big news, since it would refute Bell's theorem!

Not in any serious way. Physicists routinely assume things like measurability and continuity, etc., in their theories, and whatever results they prove don't actually hold without these assumptions, which are seldom made explicit.

In a brief Google search, I didn't see Pitowsky's original paper, but his spin-1/2 models are discussed here:
http://arxiv.org/pdf/1212.0110.pdf

 

[DrChinese]

I'd even be willing to bet real money that this isn't right -- that is, that there's no genuine example of a local theory sharing QM's predictions here. If it were true, it would indeed be big news, since it would refute Bell's theorem! (Something that many many people have wrongly claimed to do, incidentally...) But internet bets don't usually end well -- more precisely, they don't usually end at all, because nobody will ever concede that they were wrong. So instead I'll just say this: you provide a link to the paper, and I'll try to find time to take a look at it and find the mistake.

Relational BlockWorld is local. I consider it non-realistic.

http://arxiv.org/abs/quant-ph/0605105
http://arxiv.org/abs/0908.4348

How much were we betting?

 

[ttn]

I don't think he actually gave a definition of "locality". The way I interpreted what he was doing was describing a class of models, and then proving that no model in that class could reproduce the predictions of quantum mechanics. If he gave an explicit definition of what "local" means, I didn't see one.

Well I must have explained about 30 times here where you can find his careful and explicit formulation of the concept of locality, i.e., local causality.

Maybe it would help the discussion if you wrote down what you consider Bell's definition of "local".

I wrote a whole paper about it, published recently in AmJPhys. Preprint here:

http://arxiv.org/abs/0909.4553

Or see Bells' papers, especially "la nouvelle cuisine" or "the theory of local beables".

What I have seen is this: [...]

You're behind the times then. That's a standard textbook-ish sort of presentation. Bell was much better. See the above, or the systematic encyclopedia article:

http://www.scholarpedia.org/article/Bell's_theorem

which discusses all of the subtleties in gory, exhausting detail.

Pitowky showed that if you don't assume measurability of \lambda, then the EPR correlations can be explained in terms of a non-measurable function F(\hat{r}) where \hat{r} is a unit vector (or alternatively, a point on the unit sphere), with the properties that:
(This is from memory, so I might be screwing these up):

• F(\hat{r}) is always either +1 or -1.
• \langle F \rangle = \frac{1}{2}: The expectation value, over all possible values of \hat{r}, of F(\hat{r}) is 0.
• If \hat{r_1} is held fixed, and \hat{r_2} is randomly chosen so that the angle between \hat{r_1} and \hat{r_2} is \theta, then the probability that F(\hat{r_1}) = F(\hat{r_2}) is cos^2(\dfrac{\theta}{2})

Mathematically, you can prove that such functions exist (with the notion of "probability" in the above being flat lebesque measure on the set of possibilities). Pitowksy called it a "spin-1/2 function".But it's not a very natural function, and is not likely to be physically relevant.

I don't understand what measureability of anything has to do with this. It sounds like the claim is just that each particle carries local deterministic hidden variables. Such a model can account for the perfect correlations when a=b just fine of course, but cannot reproduce the general QM predictions.

Physicists routinely assume things like measurability and continuity, etc., in their theories, and whatever results they prove don't actually hold without these assumptions, which are seldom made explicit.

That is true, which is why I'm at least open to the possibility that such an assumption got made somewhere important. But so far I'm not seeing it.

In a brief Google search, I didn't see Pitowsky's original paper, but his spin-1/2 models are discussed here:
http://arxiv.org/pdf/1212.0110.pdf

Well, OK, I'll try to take a look later.

 

[stevendaryl]

You're behind the times then. That's a standard textbook-ish sort of presentation. Bell was much better.

It's from Bell, "Locality in quantum mechanics: reply to critics" in Speakable and unspeakable in quantum mechanics

I don't understand what measureability of anything has to do with this.

It's just a technical result that if you don't assume anything about measurability, it is possible to come up with a counterexample to Bell's theorem.

It sounds like the claim is just that each particle carries local deterministic hidden variables.

It is. It's exactly the type of model that Bell claimed did not exist. I don't really consider it to be a refutation of Bell's theorem, it just means that Bell's theorem should really be stated in a slightly different way, making the assumption about measurability explicit. Not that anyone really cares, because the Pitowsky model is of more mathematical than physical interest.

 

[stevendaryl]

Or see Bells' papers, especially "la nouvelle cuisine" or "the theory of local beables".

I've read his "Theory of local beables", and it seems to me that he is defining a theory of "local beables", rather than defining locality. You can fail to have local beables either by jettisoning the "local", or jettisoning the "beables".

 

[ttn]

Relational BlockWorld is local. I consider it non-realistic.

http://arxiv.org/abs/quant-ph/0605105
http://arxiv.org/abs/0908.4348

How much were we betting?

I've never even heard of "relational blockworld". I looked at one of the papers and couldn't make any sense of it -- it's just page after page of philosophy, metaphor, what the theory *doesn't* say, etc. So... you'll have to explain to me how it explains the EPR correlations -- in particular the perfect correlations when a=b. Recall that the explanation should be local (and that the "no conspiracies" assumption should be respected... something tells me this could be an issue in a "blockworld" interpretation...).

 

[ttn]

It's from Bell, "Locality in quantum mechanics: reply to critics" in Speakable and unspeakable in quantum mechanics

The point is that your'e jumping in mid-stream -- as if determinism was assumed, etc. See Bell's *full presentation* of the theorem, not some out of context snippet.

It's just a technical result that if you don't assume anything about measurability, it is possible to come up with a counterexample to Bell's theorem.

I get that that's the claim. But I'm not buying it yet.

It is. It's exactly the type of model that Bell claimed did not exist. I don't really consider it to be a refutation of Bell's theorem, it just means that Bell's theorem should really be stated in a slightly different way, making the assumption about measurability explicit. Not that anyone really cares, because the Pitowsky model is of more mathematical than physical interest.

Assuming a model of this sort actually does what you claim, I would agree. But I remain highly skeptical. Surely you are aware that all kinds of weird people (Joy Christian, for example... Hess and Phillip was another recent example) make wholly wrong claims of just this sort. Sometimes their mistakes are trivial/obvious. Sometimes they are hard to identify, for me at least. But in my experience (which is significant on this front) all of these kinds of claims always turn out to be wrong. Nevertheless, I've never heard of the one you're talking about here, and it's interesting enough to look into.

 

[ttn]

I've read his "Theory of local beables", and it seems to me that he is defining a theory of "local beables", rather than defining locality.

No, actually he's just defining locality. Look at it again. But "la nouvelle cuisine" is better. Note that he subtly tweaked how he formulated "locality" in between those papers. (See the footnote in "free variables and local causality" for some comments about why he made the change.)

You can fail to have local beables either by jettisoning the "local", or jettisoning the "beables".

So, you think a theory without beables could be local -- or for that matter nonlocal? I disagree. So did Bell: "lt is in terms of local beables that we can hope to formulate some notion of local causality." That is, without beables (i.e., physically real stuff of some kind) the very idea of locality (which is a speed limit on the influences propagating around in the stuff) is incoherent/meaningless.

 

[stevendaryl]

Relational BlockWorld is local. I consider it non-realistic.

http://arxiv.org/abs/quant-ph/0605105
http://arxiv.org/abs/0908.4348

How much were we betting?

The papers on "Relational Block World" are very frustrating, because they don't give a succinct definition of what the "Blockworld interpretation of quantum mechanics" is. The entire paper reads like a very lengthy introduction.

The observation that the generators of boosts, translations and rotations obey commutation relations isomorphic to those of quantum mechanics is intriguing (and I've wondered for years whether there was some connection), but I still don't get it. For one thing, the classical commutation relations don't involve h-bar, so I don't understand how that constant can arise from a block world interpretation (even though I don't really know what the blockworld interpretation is).

 

[ttn]

The papers on "Relational Block World" are very frustrating, because they don't give a succinct definition of what the "Blockworld interpretation of quantum mechanics" is. The entire paper reads like a very lengthy introduction.

The observation that the generators of boosts, translations and rotations obey commutation relations isomorphic to those of quantum mechanics is intriguing (and I've wondered for years whether there was some connection), but I still don't get it. For one thing, the classical commutation relations don't involve h-bar, so I don't understand how that constant can arise from a block world interpretation (even though I don't really know what the blockworld interpretation is).

There are a lot of crazy ideas for how to understand QM, and most of them simply do not make any sense. For me a useful rough litmus test is to ask the proponent of some such idea to explain what's going on in the 2-slit experiment with single electrons. Lots of theories can pass this test (Copenhagen, Bohm, MWI, GRW, for example). Ones that can't, I find I have no use for. Hopefully Dr C can give this sort of quick explanation of what this RBW thing is all about. Of course, something like this is inherent in the "challenge" I posed...

 

[bohm2]

Two other interesting papers discussing Bell's concept of local causality and implications of violation of bell's inequality pursuing Bell's and ttn's positions (with many passages from Bell's work) are the following 2 papers by M.P. Seevinck:

The starting point of the present paper is Bell’s notion of local causality and his own sharpening of it so as to provide for mathematical formalisation. Starting with Norsen’s (2007, 2009) analysis of this formalisation, it is subjected to a critique that reveals two crucial aspects that have so far not been properly taken into account. These are (i) the correct understanding of the notions of sufficiency, completeness and redundancy involved; and (ii) the fact that the apparatus settings and measurement outcomes have very different theoretical roles in the candidate theories under study. Both aspects are not adequately incorporated in the standard formalisation, and we will therefore do so. The upshot of our analysis is a more detailed, sharp and clean mathematical expression of the condition of local causality. A preliminary analysis of the repercussions of our proposal shows that it is able to locate exactly where and how the notions of locality and causality are involved in
formalising Bell’s condition of local causality.

Not throwing out the baby with the bathwater: Bell’s condition of local causality mathematically ‘sharp and clean’
http://mpseevinck.ruhosting.nl/seevinck/Bell_LC_final_Seevinck_corrected.pdf

Consider jointly the following two theorems: firstly, the so-called No-Signalling Theorem in quantum theory; and, secondly, Bell’s Theorem stating that quantum theory is not locally causal. Then, do quantum theory and the theory of (special) relativity indeed “peacefully coexist” or is there an “apparent incompatibility” here (J.S. Bell, 1984 [5, p. 172])? If we think the latter is the case—which we will argue one should—, does this ask for a radical revision of our understanding of what (special) relativity in fact enforces?

Can quantum theory and special relativity peacefully coexist?
http://mpseevinck.ruhosting.nl/seevinck/Polkinghorne_white_paper_Seevinck_Revised3.pdf

 

[harrylin]

[...] it is rather Dr C who totally misunderstands the issue. There is absolutely no *assumption* of (what Dr C means by) "realism" in Bell's 1964 paper. And Bell makes this even clearer in his many subsequent papers.

While I also don't fully agree with DrChinese, you seem to claim that Bell's referral to the "nature of reality" doesn't relate to realism at all. Sorry that doesn't make any sense to me.

The relevant money quote here is something I partially quoted earlier in this thread, from the B's sox paper: "It is remarkably difficult to get this point across, that determinism [aka, DrC's "realism"] is not a *presupposition* of the analysis. [..]

Determinism should not be confounded with realism. Bells' equation to which DrC referred imposes the particular restrained form of realism that was discussed there - not determinism. Counterfactual definiteness isn't the same as determinism.

[..] They simply *miss* that the argument begins with the EPR argument *from locality to* "realism". They look only at the *second* part of the argument, which shows that "realism" + locality implies a contradiction with experiment. So they *mistakenly* think that we get to choose which of "realism" or locality to reject, in order to avoid the conflict. But in fact there is no such choice. Locality already entails "realism". So to have to choose one to reject is to have to choose locality. [..]

That's an interesting take! However, Bell starts in 1964 with exactly the approach that you say to be a misunderstanding of Bell: "These additional variables were to restore to the theory causality and locality". And your argument doesn't seem to relate to the issue that I discovered there (after everyone else).

I think the sock is actually on the other foot.

I also think that the solution of the puzzle is likely in correcting the question (as so often).

 

[harrylin]

I'm not referring to Jaynes. I've searched the forums, but unfortunately i there's too much results for me to look through. Can you point me to the thread you are referring to? [..]

For example https://www.physicsforums.com/showthread.php?t=581193

This really isn't the place for a big technical discussion of this kind of thing. [..]

Physicsforums is THE place for big technical discussions of this kind of thing.

 

[ttn]

While I also don't fully agree with DrChinese, you seem to claim that Bell's referral to the "nature of reality" doesn't relate to realism at all. Sorry that doesn't make any sense to me.

As I have said about 30 times, "realism" is used to mean a number of different things. If I understand correctly, you are referring to the title of Bell's "Bertlmann's socks and the nature of reality". The "nature of reality" part clearly refers to the question of whether reality (i.e., nature) is local or not, which is what the paper is about. It is really not possible to talk about the question of locality vs nonlocality without talking about reality in this generalized sense. Surely there would be no point having a debate about whether unreality was local or nonlocal (i.e., whether the non-existing causal influences that propagate around between non-existing parts of a non-existing universe do or do not obey relativity's speed limit). But the point is that *this* sort of "realism" -- believing that there is a real physical world out there with causal influences propagating around in it -- is simply *not* the "realism" that (at least) DrC has in mind when he says Bell's theorem refutes realism. Is this really so hard to understand?

Determinism should not be confounded with realism. Bells' equation to which DrC referred imposes the particular restrained form of realism that was discussed there - not determinism. Counterfactual definiteness isn't the same as determinism.

The "realism" in question here means, specifically, deterministic non-contextual hidden variables. This *is* precisely what is assumed if one just jumps in and says:

"Suppose the particles carry hidden variables λ that determine the outcome for any possible measurement, so that functions A(a,λ) and B(b,λ) exist."

That is absolutely just exactly what DrC and others mean by "realism" and it is exactly what Bell was referring to when he said people mistakenly thought the argument started here instead of earlier, with a *derivation* of this "realism" from locality.

That's an interesting take! However, Bell starts in 1964 with exactly the approach that you say to be a misunderstanding of Bell: "These additional variables were to restore to the theory causality and locality". And your argument doesn't seem to relate to the issue that I discovered there (after everyone else).

So you think Bell was lying when he said people missed the first part of the argument? This is more plausible to you than the possibility that you, too, missed the first part of the argument? Give me a break. Incidentally, you have to actually read the *words* and *think* -- not just skip to the equations.

As to the issue that you "discovered there", do you mean DrC's old saw about "c" being a third angle? This is a complete and total misunderstanding on his part. First off, "c" is an *angle*, not a property or hidden variable or any such thing. To say there are three possible angles along which somebody might orient their SG device, is hardly to commit to anything like "realism". And even if what is meant is not "c" itself but the pre-determined value "along c", i.e., A(c,λ), then still -- why in the world would somebody only object when a *third* angle is introduced? Surely introducing even a *single* one -- the pre-determined value A(a,λ) -- already goes against orthodoxy by adding a pre-determined value that is nowhere to be found in QM. And if it's counter-factual definiteness that somebody is worried about, then surely introducing a *second* such pre-determined value -- the value A(b,λ) -- already commits this sin. So -- anybody who thinks that, with respect to "realism", everything is fine (i.e., no such assumption has been made) until that *third* angle "c" gets introduced, simply doesn't know what they're talking about.

That's all I'll say about this, because it's been covered ad nauseum before. If my repeated explanations here, and my invitations to go learn about all these issues from the scholarpedia article, don't make you understand then nothing ever will.

 

[ttn]

Physicsforums is THE place for big technical discussions of this kind of thing.

Maybe so, but not this thread. Let's just say that the set of people who voted "anti-realism" in the poll *because of some issue having to do with the measureability of the space of hidden variable states* is almost certainly of measure zero.

Look, there are two categories of issues here. First, there are the "elementary issues" such as what you raised in your previous post. For example: do you understand that, contrary to how most textbooks present it, Bell's theorem does *not* simply begin with the *assumption* of deterministic non-contextual hidden variables, but instead begins by *deriving* these from locality and the perfect correlations? Do you understand that therefore you cannot avoid the conclusion of nonlocality by denying determinism or hidden variables or non-contextuality or counter-factual definiteness? Do you understand that there is also a "no conspiracies" assumption that is made in proving the theorem? Do you know where this comes in (already in the first, EPR part!) and do you understand that it has nothing to do with literal human freewill? Do you understand that the "locality" from which the inequality flows is *not* defined by some statement like A(a,b,λ) = A(a,λ) but is rather something that Bell gave an extremely careful, general, non-proprietary formulation of?

People suffering from confusions about issues like these simply need to go and read a bit more until they understand the issues.

Now, admittedly, there are also "advanced issues", some of which have come out in this thread. For example, isn't there a kind of inconsistency between the "no conspiracy" assumption and Bell's formulation of locality? Doesn't Bell tacitly assume that the space of physical states λ is measureable, in deriving the inequality? And: doesn't Bell's formulation of locality become somewhat difficult to apply to theories whose ontologies include nonlocal beables?

That is, there are legitimate and difficult and controversial questions about this stuff. But, seriously, how many people voted for "anti-realism" because of anything like this? The answer is: one or two at most. The rest voted for "anti-realism" because they are simply confused (like DrC) about elementary points. My goal here has been to try to help make people aware that they might be confused. This is admittedly sometimes hard to figure out, since lots of seemingly reputable people, even books, are confused in exactly the same ways. So here is my final plea. If you are somebody who voted for "anti-realism", please simply dismiss/ignore everything I'm saying -- if I strike you as somebody who doesn't know what he's talking about, who hasn't read and understood Bell, who hasn't thought seriously and carefully about these issues, etc. That is, if you think I'm a crackpot, then just ignore me. On the other hand, if you get the impression that I have studied these issues carefully, that I do seem to know something about what I'm talking about, etc. -- then take it seriously that I am saying YOU ARE CONFUSED. Go read some of the stuff I've been linking to, so that next time there's a poll like this, we don't have to again witness the embarrassing spectacle that we have witnessed here!

 

[harrylin]

[..] The "nature of reality" part clearly refers to the question of whether reality (i.e., nature) is local or not, which is what the paper is about. [..] But the point is that *this* sort of "realism" -- believing that there is a real physical world out there with causal influences propagating around in it -- is simply *not* the "realism" that (at least) DrC has in mind when he says Bell's theorem refutes realism. Is this really so hard to understand? [..]

We agree about that but apparently you didn't understand that; sorry if you somehow ascribed to me something that I disagree with. What I referred to is the facts of Bell's derivation that DrChinese pointed out in the other thread; Bell's derivation is not subject to DrChinese's interpretation of "realism".

"Suppose the particles carry hidden variables λ that determine the outcome for any possible measurement, so that functions A(a,λ) and B(b,λ) exist."

That is absolutely just exactly what DrC and others mean by "realism" and it is exactly what Bell was referring to when he said people mistakenly thought the argument started here instead of earlier, with a *derivation* of this "realism" from locality.

You seem to be beating a strawman and I'm not interested in that.

So you think Bell was lying when he said people missed the first part of the argument?

No, see above...

As to the issue that you "discovered there", do you mean DrC's old saw about "c" being a third angle? [..]

Once more no, see above, I only referred to Bell's derivation.

 

[stevendaryl]

Maybe so, but not this thread. Let's just say that the set of people who voted "anti-realism" in the poll *because of some issue having to do with the measureability of the space of hidden variable states* is almost certainly of measure zero.

I certainly agree with that.

Look, there are two categories of issues here. First, there are the "elementary issues" such as what you raised in your previous post. For example: do you understand that, contrary to how most textbooks present it, Bell's theorem does *not* simply begin with the *assumption* of deterministic non-contextual hidden variables, but instead begins by *deriving* these from locality and the perfect correlations?

I don't believe that you are right about that. If you're claiming that Bell's "Theory of Local Beables" is his definition of locality, then he's already assumed essentially that hidden variables exist. He hasn't derived it from locality, because he doesn't even offer a way to state locality in the absence of such hidden variables.

 

[harrylin]

Maybe so, but not this thread.

Ah yes, due to the title of this thread I had forgotten that it was just meant for an opinion poll!

[..] Bell's theorem does *not* simply begin with the *assumption* of deterministic non-contextual hidden variables, but instead begins by *deriving* these from locality and the perfect correlations? Do you understand that therefore you cannot avoid the conclusion of nonlocality by denying determinism or hidden variables or non-contextuality or counter-factual definiteness? [..]

You are here summarizing a claim (with "therefore") that I have not seen discussed on this forum; as you said, this is not the thread for elaborating on such things. It would make for an interesting thread on itself!

how many people voted for "anti-realism" because of anything like this? The answer is: one or two at most. The rest voted for "anti-realism" because they are simply confused (like DrC) about elementary points. My goal here has been to try to help make people aware that they might be confused. This is admittedly sometimes hard to figure out, since lots of seemingly reputable people, even books, are confused in exactly the same ways. [..]

The main problem (which I think is often recognized) is that there are too different (disagreeing) understandings about the meaning of words. Consequently such opinion polls can never be more than an indication of along which lines people are currently thinking.

 

[stevendaryl]

I don't believe that you are right about that. If you're claiming that Bell's "Theory of Local Beables" is his definition of locality, then he's already assumed essentially that hidden variables exist. He hasn't derived it from locality, because he doesn't even offer a way to state locality in the absence of such hidden variables.

Let me describe a toy model of EPR measurements that I think illustrates that it is possible to have locality without realism, so locality doesn't imply realism.

We have Alice at her detector, and far away, we have Bob at his detector. They each do the following things, over and over:

1. Pick a detector orientation.
2. Measure the spin of one of the particles from a twin pair source for that orientation.
3. Record the results and the detector orientation on a piece of paper.
4. Send the results in a letter to the other experimenter.

Here's the twist in the story: Alice and Bob both have terrible handwriting and/or terrible vision. So when Alice writes "I measured spin up along the z-axis", Bob sometimes reads it to say "I measured spin down along the z-axis", and vice-verse. Similarly, Alice occasionally misinterprets what Bob wrote.

If we further assume that the probability of a misinterpretation depends on (A) what was actually written, and (B) the state of the experimenter doing the reading, then it is certainly possible to reproduce the EPR results without faster-than-light influences.

This resolution does not deny locality, it denies realism, in that it doesn't assume that the words "Alice measured spin-up along the z-axis" is a reliable record of anything real in the world.

This is not a serious suggestion as to what is going on in quantum mechanics, but just a demonstration that no single experimental result, such as the EPR result, can be taken to show nonlocality, without additional realism assumptions.

 

[ttn]

I don't believe that you are right about that. If you're claiming that Bell's "Theory of Local Beables" is his definition of locality, then he's already assumed essentially that hidden variables exist. He hasn't derived it from locality, because he doesn't even offer a way to state locality in the absence of such hidden variables.

I believe you must be thinking that "beables" is synonymous with "hidden variables"? That is not the case. Bell is super careful and explains his terminology. "Beables" means "whatever a certain candidate theory says one should take seriously, as corresponding to something that is physically real." (Those are my words, but it's his idea.) He gives examples like the E and B fields in classical electromagnetism, as against the potentials V and A. He also applies his formulation to ordinary QM, in which (at most!) the wave function is the only beable in the picture for microscopic things. So it simply is not the case that "beable", as Bell uses the term, means the same as "hidden variable". If you think that, you're confused and should go back and read Bell again.

 

[ttn]

We agree about that but apparently you didn't understand that; sorry if you somehow ascribed to me something that I disagree with. What I referred to is the facts of Bell's derivation that DrChinese pointed out in the other thread; Bell's derivation is not subject to DrChinese's interpretation of "realism".

Perhaps I misunderstood you to be agreeing with DrC's criticism, where in fact you were pointing to the place you thought you had refuted his criticism? If so, I'm sorry for my misunderstanding. But in any case, I'm tired of arguing this point and so will simply leave it at that. I've made my view clear, or at least as clear as I know how to do, and have given references for people who want to think about it more.

 

[stevendaryl]

I believe you must be thinking that "beables" is synonymous with "hidden variables"? That is not the case.

How is it different from hidden variables?

Bell is super careful and explains his terminology. "Beables" means "whatever a certain candidate theory says one should take seriously, as corresponding to something that is physically real."

But does quantum mechanics necessarily have such "beables"? I don't see that it does. A candidate such as "the electric field" doesn't work as such a beable, because the electric field depends on locations of charged particles, and quantum mechanics doesn't assume that particles have definite positions. So what's an example of a beable in quantum mechanics?

 

[ttn]

Let me describe a toy model of EPR measurements that I think illustrates that it is possible to have locality without realism, so locality doesn't imply realism.

We have Alice at her detector, and far away, we have Bob at his detector. They each do the following things, over and over:

1. Pick a detector orientation.
2. Measure the spin of one of the particles from a twin pair source for that orientation.
3. Record the results and the detector orientation on a piece of paper.
4. Send the results in a letter to the other experimenter.

Here's the twist in the story: Alice and Bob both have terrible handwriting and/or terrible vision. So when Alice writes "I measured spin up along the z-axis", Bob sometimes reads it to say "I measured spin down along the z-axis", and vice-verse. Similarly, Alice occasionally misinterprets what Bob wrote.

If we further assume that the probability of a misinterpretation depends on (A) what was actually written, and (B) the state of the experimenter doing the reading, then it is certainly possible to reproduce the EPR results without faster-than-light influences.

This resolution does not deny locality, it denies realism, in that it doesn't assume that the words "Alice measured spin-up along the z-axis" is a reliable record of anything real in the world.

This is not a serious suggestion as to what is going on in quantum mechanics, but just a demonstration that no single experimental result, such as the EPR result, can be taken to show nonlocality, without additional realism assumptions.

Well, thank you for at least *attempting* to address the challenge! But I don't think this really does it. What we are looking for is a local-but-non-realist explanation of the actual results that the experiment will give (in the special case where a=b, or equivalently for those particle pairs for which a happens to = b). My claim is that, if the underlying *physics* model is local and non-realist, then it will predict that at least sometimes, when a=b, the results will not be perfectly correlated. (For example, if we are calibrating things such that QM says the results should always be the *same* when a=b, then this local-but-non-realist model will say that, at least sometimes, the results will be *different* when a=b.) Now undoubtedly you can cook up a goofy story like the one here in which a sequence of conspiratorial accidents and misinterpretations fools Alice and Bob into believing that the experiment exhibited perfect correlations. But come on, that's not serious. What we are interested in is explain the actual, QM-predicted and experiment-verified correlations... not cooking up a "perfect correlations" delusion in the mind of some imaginary person.

Also, what you said above about the sense in which the proposed model is non-realist makes no sense. At best, this is yet another distinct sense of "realism". But it has nothing to do with the deterministic non-contextual hidden variables sense of "realism" that DrC and others who voted "anti-realism" in the poll think is relevant here.

 

[ttn]

How is it different from hidden variables?

See my "Bell's concept of local causality" paper where there is a whole section discussing this with copious quotes from Bell.

But does quantum mechanics necessarily have such "beables"? I don't see that it does.

You can say it doesn't if you want. But that theory will still be nonlocal.

A candidate such as "the electric field" doesn't work as such a beable, because the electric field depends on locations of charged particles, and quantum mechanics doesn't assume that particles have definite positions. So what's an example of a beable in quantum mechanics?

The wave function, at least for people who think (following Bohr) that the wave function provides a complete description of (microscopic) physical reality. It's true, there are people who don't think the wf in ordinary QM should be understood as a beable, as corresponding to some physical reality. The question for them is: what, then, does? Let them specify what their theory is. If their theory is ordinary QM -- but with *no beables* for the microscopic world -- OK, that's a perfectly clear theory, but it is *really* nonlocal since in effect it has direct, unmediated causal influences between spacelike separated hunks of measuring equipment.

But... seriously... take some time to look into Bell's formulation of locality before you keep coming up with all these alleged objections to it. Note that it is also a mistake to think that one will learn too much by scrutinizing an "unprofessionally vague and ambiguous" theory like orthodox QM...

 

[stevendaryl]

Well, thank you for at least *attempting* to address the challenge! But I don't think this really does it. What we are looking for is a local-but-non-realist explanation of the actual results that the experiment will give (in the special case where a=b, or equivalently for those particle pairs for which a happens to = b).

Well, I can certainly stipulate that

If Bob measured the spin along axis \hat{b} and got result R_b (either +1 or -1), and the message from Alice said "I measured the spin along axis \hat{a} and got result [blah, blah, blah].", then Bob will read the "blah, blah, blah" as equal to R_b with probability sin^2(\dfrac{\theta}{2}).​

I agree that this is a bizarre, comical way of resolving the conundrum, but it's got the same flavor as theories such as MWI that deny that measurements have definite values. That's why, in spite of your insistence that Bell's theorem is only about locality, I insist that some kind of realism assumption is required to derive nonlocality.

My claim is that, if the underlying *physics* model is local and non-realist, then it will predict that at least sometimes, when a=b, the results will not be perfectly correlated.

Why? We can specify that Bob's probability of misreading Alice's message depends on Bob's state, as well as the state of Alice's message. That's perfectly local. We can certainly make our probabilities such that it becomes a certainty in certain circumstances.

Also, what you said above about the sense in which the proposed model is non-realist makes no sense.

Maybe some other word than "realist" is called for, but the point is that Bob will be constructing a history of what happened based on his reading of the messages from Alice, but that history does not reflect anything real (at least as far as the parts referring to Alice's results).

At best, this is yet another distinct sense of "realism". But it has nothing to do with the deterministic non-contextual hidden variables sense of "realism" that DrC and others who voted "anti-realism" in the poll think is relevant here.

That's not clear to me.

 

[stevendaryl]

You can say it doesn't if you want. But that theory will still be nonlocal.

Well, that's just a terminology thing. I don't agree with that terminology. I think it's misleading.

The wave function, at least for people who think (following Bohr) that the wave function provides a complete description of (microscopic) physical reality.

I don't think that there is general agreement that the wave function is objectively real. According to Everett's definitions, the wave function of an electron (say) is relative to the observer.

It's true, there are people who don't think the wf in ordinary QM should be understood as a beable, as corresponding to some physical reality. The question for them is: what, then, does?

That's a very good question, and the answer seems to be: we don't know.

 

[ttn]

Well, I can certainly stipulate that
[...]

Heck, if you're going to just stipulate stuff, why not just stipulate the existence of a local theory that explains all the QM predictions?

I agree that this is a bizarre, comical way of resolving the conundrum, but it's got the same flavor as theories such as MWI that deny that measurements have definite values. That's why, in spite of your insistence that Bell's theorem is only about locality, I insist that some kind of realism assumption is required to derive nonlocality.

More seriously, I agree with you here -- both about its being bizarre / comical / unserious and about its being substantially similar to MWI. (Yes, that was supposed to be funny, but I actually mean it, too.)

I'm trying to gradually extract myself from this thread, so the last thing I want to do here is get into a big side discussion of MWI. But one of the crucial points is what you more or less said here: if "explaining the QM predictions locally" means explaining what Aspect et al. think actually occurred in their lab, then there's a really important sense in which MWI doesn't do this at all. It says that, actually, something quite radically different happened, than what Aspect et al though. And then it tells an elaborate fairy tale about how, nevertheless, it predicts that Aspect et al should be deluded into thinking what they thought. That is, instead of explaining what one (perhaps naively) thinks needs explaining, it instead (allegedly) explains how the subjective delusion (that the outcomes predicted by QM actually happened) arises in consciousness. Whatever else anybody wants to say about it, that's ... not the same thing.

And second, I think it is highly dubious to say that MWI is a local theory. It's not clear what the local beables are supposed to be, and I stand with Bell in thinking that theories without local beables certainly cannot be meaningfully asserted to be local, or even nonlocal. It's like calling beethoven's 5th symphony local. There is at least one attempt I know of to be clear and explicit about local beables for MWI, but on that version actually it turns out that the theory is nonlocal.

http://arxiv.org/abs/0903.2211

Why? We can specify that Bob's probability of misreading Alice's message depends on Bob's state, as well as the state of Alice's message. That's perfectly local. We can certainly make our probabilities such that it becomes a certainty in certain circumstances.

I'm sorry, but as soon as you start talking about Bob's misreading of Alice's message -- instead of what Alice's actual outcome was -- I lose interest.

Maybe some other word than "realist" is called for, but the point is that Bob will be constructing a history of what happened based on his reading of the messages from Alice, but that history does not reflect anything real (at least as far as the parts referring to Alice's results).

Yes, I get that that's what you have in mind, and you're absolutely right that it's quite relevant to MWI. But surely you can see how it's a form of simply "cheating" to play this kind of game. I don't mean that such ideas are necessarily not worth considering (though personally I find them rather silly). But it is *really* changing the underlying rules of the discussion when instead of explaining the facts, you say that everybody is deluded about the facts and start trying to explain the delusions. See how far that kind of game gets you in other fields in science, for example: "Actually my design for the bridge *was* good -- you are just deluded into thinking that it collapsed and killed all those people."

That's not clear to me.

People think that, to derive a Bell inequality, you need several assumptions including at least (a) locality and (b) deterministic non-contextual hidden variables. Let's call (b) "realism" for short. People then see that experiments violate the inequality. (Note here they are not at all thinking "ooh, maybe we are all only *deluded* into thinking the inequality is violated, because we are actually *deluded* about any of the individual measurements having had any definite outcome at all!". That thought doesn't enter a normal person's mind! They take the data at face value.) So they say we have to reject (a) or (b). They say that it's crazy to reject (a) since (a) is required by relativity. Whereas, they say, only senile old fools like Einstein ever believed in (b), and indeed there are a bunch of no-go theorems basically providing independent proof that we shouldn't believe in (b), so, they say, the choice is obvious: bell's theorem shows that we should reject (b).

People who voted "anti-realism" in the poll are of course free to explain their reasoning if this doesn't capture it, but I'm pretty sure that's the main idea for most of these people.

 

[ttn]

Well, that's just a terminology thing. I don't agree with that terminology. I think it's misleading.

Well fine, but it's not as if it's arbitrary or undefined terminology. I've explained repeatedly exactly what I mean by "locality", referring to Bell's formulation, etc. So if you want to use the word "local" to mean something distinct, that's of course no problem. But don't mistake doing that for constructing an argument that Bell's formulation somehow fails to capture the concept it's intended to capture.

I don't think that there is general agreement that the wave function is objectively real. According to Everett's definitions, the wave function of an electron (say) is relative to the observer.

Of course there's not general agreement. But the point is that it doesn't matter. There's a theory -- let's call it QM1 -- which is orthodox QM with the wf interpreted as a beable. That theory is nonlocal. Then there's another theory -- let's call it QM2 -- which is orthodox QM with the wf interpreted as *not* a beable. That theory is nonlocal.

That's a very good question, and the answer seems to be: we don't know.

And neither do they. That is, maybe somebody will come up with a new candidate account of what the beables are. That will be a new theory. Perhaps it will be a local theory. If so, then it will make empirical predictions that respect Bell's inequality. That's what the theorem says. Note that we don't have to wait around for the people to actually to come up with their theory (or figure out how they think QM should be interpreted, or anything like that) in order to know this. That's the beauty of the theorem.

 

[stevendaryl]

Heck, if you're going to just stipulate stuff, why not just stipulate the existence of a local theory that explains all the QM predictions?

My model is such a theory of EPR-type predictions. So I don't need to stipulate its existence.

if "explaining the QM predictions locally" means explaining what Aspect et al. think actually occurred in their lab, then there's a really important sense in which MWI doesn't do this at all. It says that, actually, something quite radically different happened, than what Aspect et al though. And then it tells an elaborate fairy tale about how, nevertheless, it predicts that Aspect et al should be deluded into thinking what they thought.

That's what, to me, the non-realist branch of the "nonlocal or nonrealistic" choice means--that the world is VERY different from what our senses would lead us to expect.

And second, I think it is highly dubious to say that MWI is a local theory. It's not clear what the local beables are supposed to be,

Well, one possibility, not for quantum mechanics, but for quantum field theory, is to take as the "beables" not the wave function, but the field operators. They obey a perfectly local evolution equation.

I'm sorry, but as soon as you start talking about Bob's misreading of Alice's message -- instead of what Alice's actual outcome was -- I lose interest.

Well, in that case, you should rephrase your claims about what Bell's theorem shows as: "Among the theories that I am interested in, the only ones consistent with quantum mechanics are nonlocal".

 

[stevendaryl]

Of course there's not general agreement. But the point is that it doesn't matter. There's a theory -- let's call it QM1 -- which is orthodox QM with the wf interpreted as a beable. That theory is nonlocal. Then there's another theory -- let's call it QM2 -- which is orthodox QM with the wf interpreted as *not* a beable. That theory is nonlocal.

QM2 is not a theory of "beables" at all, local or otherwise.

 

[stevendaryl]

(Note here they are not at all thinking "ooh, maybe we are all only *deluded* into thinking the inequality is violated, because we are actually *deluded* about any of the individual measurements having had any definite outcome at all!". That thought doesn't enter a normal person's mind! They take the data at face value.)

Well, what normal people would believe is not that relevant. Normal people don't care about quantum mechanics, one way or the other.

It seems to me that if you are willing to accept that an electron can be in a superposition of states, then there is no principled reason to reject the possibility of a person, or a galaxy being in a superposition of states. So if you reject the latter possibility out of hand, then it means that you are already assuming that there is some unknown fact of the matter about questions like "where is the electron right now?"

So it appears to me that you are assuming from the start that there is some hidden variables underlying quantum mechanics, and the only question is whether its local or nonlocal. I agree, if you have a hidden variables model, it has to be nonlocal. But the people who take the "nonrealist" branch of the question "nonrealist or nonlocal" are rejecting the existence of hidden variables.

 

[DrChinese]

I've never even heard of "relational blockworld". I looked at one of the papers and couldn't make any sense of it -- it's just page after page of philosophy, metaphor, what the theory *doesn't* say, etc. So... you'll have to explain to me how it explains the EPR correlations -- in particular the perfect correlations when a=b. Recall that the explanation should be local (and that the "no conspiracies" assumption should be respected... something tells me this could be an issue in a "blockworld" interpretation...).

I'm sure you haven't heard about a lot of things.

Yet, here it is! And it is fairly well developed for such a theory. It is what I refer to as a time-symmetric class theory. A context is not limited to the past and/or present, and so that is how it is able to account locally for Bell correlations. You don't have to agree with it, and it fact it makes predictions which may prove false. But it is a working theory.

I have no doubt that you will deny the existence of this, as this would be ipso facto evidence that your main contention is incorrect. That being that Bell+Aspect implies non-locality. As I have said many times, QM+Bell implies local hidden variable theories are non-starters.

 

[stevendaryl]

Yet, here it is! And it is fairly well developed for such a theory. It is what I refer to as a time-symmetric class theory. A context is not limited to the past and/or present, and so that is how it is able to account locally for Bell correlations. You don't have to agree with it, and it fact it makes predictions which may prove false. But it is a working theory.

I looked at the papers that you gave URLs for, and, as I said, I couldn't see a succinct, precise definition of what the model is. They mention how the quantum mechanical commutation relations have a similarity to the commutation relation between various symmetry operators in relativity (translations, rotations, boosts). But that doesn't seem to be a model, it's just an observation.

 

[stevendaryl]

I looked at the papers that you gave URLs for, and, as I said, I couldn't see a succinct, precise definition of what the model is. They mention how the quantum mechanical commutation relations have a similarity to the commutation relation between various symmetry operators in relativity (translations, rotations, boosts). But that doesn't seem to be a model, it's just an observation.

I think this review article gives a better summary of the chief ideas:
http://chaos.swarthmore.edu/research/Dan.pdf

 

[rubi]

I've read your paper and some other stuff from Bell and this thread now. Let's see:

I'm sorry, but... what the heck are you talking about? Are you really saying that ordinary QM doesn't allow you to calculate what the probabilities of various possible measurement outcomes are, in terms of the state ? of the system in question? That's the one thing that orthodox QM is unquestionably, uncontroversially good for!

Maybe the issue has to do with what I assume(d) was just a typo? Namely: it's not p(A,B,\lambda) but rather p(A,B|\lambda) -- or, as I indicated before, slightly more precisely, p_{\lambda}(A,B).

I don't think that helps. If i understood your paper correctly, then p(A,B|\lambda) is a conditional probability and then \lambda needs to be an element of a probability space. Otherwise, how would you apply the rules of probability theory if your objects aren't well-defined probability measures? I think it would be a good idea to make it more clear in your paper, what the objects you are talking about mean and what spaces they belong to in terms of short, precise mathematical statements instead of rather long, vague paragraphs of text. Maybe it can be made rigorous, but at the moment i don't see it. I just see that you derive the factorization property that is used in the derivation of Bell's inequality, but in order to derive Bell's inequality, you have to perform an integration over \lambda, which isn't possible if \lambda is supposed to be the wave-function. So even if you don't want to integrate over it, it should be possible in principle. (At least from what i understood, the factorization property you derive is supposed to be the same that is used in the derivation of Bell's inequality, right?)


----


But here's another thing i noticed: If i understood it correctly, the beables of a theory are supposed to be things that are ascribed a physical reality. Then i think that in QM, the individual measurement outcomes aren't beables. Neither is the wave-function. For an advocate of the Copenhangen interpretation, the only beables of QM would be the probability distributions.

For example the fact that after a measurement, the position probability distribution is peaked over a sharp value doesn't mean that the particle has suddenly acquired the real physical property of having a definite position, albeit it didn't have it one moment before. It merely means that we have come to know more about the probability distribution itself then we did before. The same thing applies to spin. An individual measurement tells us nothing about nature. Only the totality of many measurements allows us to make a statement about the world.

Also, the wave-function is not a beable. It's just a tool that is used to calculate the probability distributions, much like the 4-potential in electrodynamics is just a tool to calculate the field strength. A Copenhagenist wouldn't ascribe physical reality to the wave-function.


If the individual measurements and the wave-function are not beables. then i think you wouldn't come across these technical difficulties about measures in infinite-dimensional spaces. However, i have not studied what the theory would assert about the locality of QM, then.

 

[DrChinese]

I looked at the papers that you gave URLs for, and, as I said, I couldn't see a succinct, precise definition of what the model is. They mention how the quantum mechanical commutation relations have a similarity to the commutation relation between various symmetry operators in relativity (translations, rotations, boosts). But that doesn't seem to be a model, it's just an observation.

It's not my theory, I won't attempt to explain it or defend it. It is serious work though. Unlike every other candidate QM theory/model/interpretation I am aware of, it goes out on a limb to make a specific prediction which is different than orthodox QM. So give 'em credit for bravery if nothing else. Time will tell if their cosmological model predictions pan out, there is a lot of active research in that particular area (dark matter).

Thanks for the link to the Peterson paper, you are correct that it offers a view which is more relevant for this thread.

 

[stevendaryl]

It's not my theory, I won't attempt to explain it or defend it. It is serious work though. Unlike every other candidate QM theory/model/interpretation I am aware of, it goes out on a limb to make a specific prediction which is different than orthodox QM. So give 'em credit for bravery if nothing else. Time will tell if their cosmological model predictions pan out, there is a lot of active research in that particular area (dark matter).

Thanks for the link to the Peterson paper, you are correct that it offers a view which is more relevant for this thread.

As I said, I don't really understand what the model really is, except for the vague idea that it assumes that relativity is at the heart of quantum indeterminacy in some way. In this respect, it reminds me of Cramer's "Transactional Interpretation", which also explains the seeming nondeterminism of quantum measurement results in terms of details that can involve future as well as past. Cramer also assumed that non-relativistic quantum mechanics contained a remnant of relativity.

 

[bohm2]

I don't think that helps. If i understood your paper correctly, then p(A,B|λ) is a conditional probability and then λ needs to be an element of a probability space. Otherwise, how would you apply the rules of probability theory if your objects aren't well-defined probability measures?

Did you read the other paper I linked above by M. P. Seevinck. If I'm understanding your question, (I might not be) I believe Seevinck alludes to that beginning in section IV: INTRODUCING MATHEMATICS: FORMALIZING SUFFICIENCY

Then how are we to mathematically implement Bell’s idea of "λ being sufficiently specified so as to declare redundant some of the conditional variables” in P_a,b(A,B|λ), where the latter are in fact to range over both the labels a, b and the random variables A,B? This we will perform next...

Not throwing out the baby with the bathwater: Bell’s condition of local causality mathematically ‘sharp and clean’
http://mpseevinck.ruhosting.nl/seevinck/Bell_LC_final_Seevinck_corrected.pdf

 

[rlduncan]

You have a green ball and a red ball and put them in two identical boxes. You send these boxes to two different people. These people know that you started with a green and a red ball. So the probability to get green/red is 1/2. When person 1 opens his box, he will get a definite result. Let's say he gets red. Then he knows immediately that person 2 has the green ball in his box, even if that box hasn't been opened yet. This is definitely a nonlocal correlation, but nobody would consider this as an action at a distance.

Let's use Alice and Bob as the two different people and calculate the probability using the conditional probability formula.

P(Alice-green,Bob-red)=(.05)(1)=0.5 and P(Alice-red,Bob-green)=(0.5)(1)=0.5 and the probability that Alice and Bob get different colors is 0.5+0.5=1. Standard calculation.

Question. How does Bell mathematically calculate using his formula for the joint probability by factoring to obtain a product of individual probabilities and incorporating λ to explain this perfect anti-correlation of opposite colors? Define λ for this case if possible and explain how it allows one to get to this same probability, P=1.

Thanks

 

[ttn]

I'm sure you haven't heard about a lot of things.

Yet, here it is! And it is fairly well developed for such a theory. It is what I refer to as a time-symmetric class theory. A context is not limited to the past and/or present, and so that is how it is able to account locally for Bell correlations. You don't have to agree with it, and it fact it makes predictions which may prove false. But it is a working theory.

I have no doubt that you will deny the existence of this, as this would be ipso facto evidence that your main contention is incorrect.

I have no doubt that "the theory exists" in the sense that people have written some papers about it, etc. But whether it is genuinely a "working theory" or not is a different question. To me it is telling that even you -- who raised it and apparently thinks it's a counterexample to my claims -- cannot or will not explain anything about how it works and in particular how it explains the perfect correlations in a local but non-realist way. Surely if you understood this you'd be bursting at the seams to prove me wrong...

As I have said many times, QM+Bell implies local hidden variable theories are non-starters.

That's true. But it also commits the fallacy of the superfluous adjective.

 

[stevendaryl]

Let's use Alice and Bob as the two different people and calculate the probability using the conditional probability formula.

P(Alice-green,Bob-red)=(.05)(1)=0.5 and P(Alice-red,Bob-green)=(0.5)(1)=0.5 and the probability that Alice and Bob get different colors is 0.5+0.5=1. Standard calculation.

Question. How does Bell mathematically calculate using his formula for the joint probability by factoring to obtain a product of individual probabilities and incorporating λ to explain this perfect anti-correlation of opposite colors? Define λ for this case if possible and explain how it allows one to get to this same probability, P=1.

Thanks

In this case, a "hidden variable" \lambda is just a specification of who gets the green ball. So there are two possible values of \lambda: A_g and B_g. So

P(Alice-green, Bob-red \vert\ \lambda = A_g) = 1
P(Alice-green, Bob-red \vert\ \lambda = B_g) = 0
P(Alice-red, Bob-red \vert\ \lambda = A_g) = 0
P(Alice-red, Bob-green\vert\ \lambda = B_g) = 1

 

[ttn]

I've read your paper and some other stuff from Bell and this thread now.

Thanks for taking the time to do that and for sharing your comments here.

I don't think that helps. If i understood your paper correctly, then p(A,B|\lambda) is a conditional probability and then \lambda needs to be an element of a probability space. Otherwise, how would you apply the rules of probability theory if your objects aren't well-defined probability measures? I think it would be a good idea to make it more clear in your paper, what the objects you are talking about mean and what spaces they belong to in terms of short, precise mathematical statements instead of rather long, vague paragraphs of text. Maybe it can be made rigorous, but at the moment i don't see it. I just see that you derive the factorization property that is used in the derivation of Bell's inequality, but in order to derive Bell's inequality, you have to perform an integration over \lambda, which isn't possible if \lambda is supposed to be the wave-function. So even if you don't want to integrate over it, it should be possible in principle.

I agree that some things in the "Bell's concept..." paper are not as mathematically precise as one could wish. This is in part because it's a pedagogical paper (for physics students and teachers) and partly because I think there is a point at which mathematical precision actually gets in the way of clear understanding. Perhaps you would like the scholarpedia "Bell's theorem" entry more -- it's a bit more technical (with two of the four authors being mathematicians) and covers the same ideas.

But as to the λ thing, it seems here is an example where the "vague paragraphs of text" are actually important to take in. The λ refers to the possible microstates of the particle pair that can be produced by a given sort of preparation procedure. For QM, the preparation procedure produces a pair in the spin singlet state, period. So ρ(λ) is a delta function! Do you think there's some problem integrating over that when the time comes?

(At least from what i understood, the factorization property you derive is supposed to be the same that is used in the derivation of Bell's inequality, right?)

Yes, absolutely.

But here's another thing i noticed: If i understood it correctly, the beables of a theory are supposed to be things that are ascribed a physical reality. Then i think that in QM, the individual measurement outcomes aren't beables. Neither is the wave-function. For an advocate of the Copenhangen interpretation, the only beables of QM would be the probability distributions.

Bohr's Copenhagen interpretation insisted that the directly perceivable macroscopic classical world existed. (He literally insisted repeatedly on this, in the context of saying that all empirical data was ultimately statements about such macroscopic things.) So strictly speaking, the Copenhagen interpretation involves dividing the world into two realms -- the classical/macro realm and the quantum/micro realm. The former just unproblematically exists, essentially by postulate, and so there are lots and lots of beables there. The beable-status of the micro-world for Copenhagen is indeed more controversial, as has already been discussed on this thread. But (as discussed) that doesn't matter. In short, the "measurement outcomes" most certainly *are* beables for Copenhagen. Bohr himself insisted on it specifically. And to deny such outcomes beable status -- for *any* theory -- is frankly borderline crazy. We are talking here about concrete things like which way a certain pointer in a certain lab pointed at a certain time. To deny "physically real" status to such things is ... well ... to approach solipsism.

For example the fact that after a measurement, the position probability distribution is peaked over a sharp value doesn't mean that the particle has suddenly acquired the real physical property of having a definite position, albeit it didn't have it one moment before. It merely means that we have come to know more about the probability distribution itself then we did before. The same thing applies to spin. An individual measurement tells us nothing about nature. Only the totality of many measurements allows us to make a statement about the world.

What exactly you say is going on physically at the micro-level of course depends on which theory you're talking about and in particular what objects have beable status for the theory in question. But I fundamentally disagree about the last part of what you write. An individual measurement absolutely does tell us something about nature. Think about what an individual measurement means, concretely and physically. It means (at least) that some macroscopic object (think: pointer) moved a certain way. That is something we can directly perceive. It is pre-eminently physical, a fact about nature. (This was also one of the points that I guess you glossed over in the "long, vague paragraphs of text".)

Also, the wave-function is not a beable. It's just a tool that is used to calculate the probability distributions, much like the 4-potential in electrodynamics is just a tool to calculate the field strength. A Copenhagenist wouldn't ascribe physical reality to the wave-function.

Some do, some don't. But (as discussed above) this doesn't matter. You can give it either status you want, and "Copenhagen QM" is still nonlocal.

 

[ttn]

I think this review article gives a better summary of the chief ideas:
http://chaos.swarthmore.edu/research/Dan.pdf

I just spent an hour with it and still have essentially no idea what the point is. Certainly there is no genuine physical theory here. Sigh.

 

[stevendaryl]

I just spent an hour with it and still have essentially no idea what the point is. Certainly there is no genuine physical theory here. Sigh.

I wouldn't say "certainly", but it's not clear to me what the point is, either. I understand the derivation that quantum commutation relations can be interpreted as commutation relations of generators of Poincare symmetries, but I don't understand what's supposed to follow from that.

 

[rubi]

I agree that some things in the "Bell's concept..." paper are not as mathematically precise as one could wish. This is in part because it's a pedagogical paper (for physics students and teachers) and partly because I think there is a point at which mathematical precision actually gets in the way of clear understanding. Perhaps you would like the scholarpedia "Bell's theorem" entry more -- it's a bit more technical (with two of the four authors being mathematicians) and covers the same ideas.

I agree that technical expositions often aren't very pedagogical. But i think that at some point, there should be some place, where the axioms of the theory are written down in a precise way and the theorems are proven rigorously. I believe that much of the power of physics comes from the fact that the mathematical language that underpins physics is very pedantic. I think it's desirable to know the limitations of our theories. This can often lead to a deeper understanding and even new discoveries.

I will take a look at the scholarpedia article.

But as to the λ thing, it seems here is an example where the "vague paragraphs of text" are actually important to take in. The λ refers to the possible microstates of the particle pair that can be produced by a given sort of preparation procedure. For QM, the preparation procedure produces a pair in the spin singlet state, period. So ρ(λ) is a delta function! Do you think there's some problem integrating over that when the time comes?

If \rho(\lambda) were a delta function, then it's possible to formulate this using the dirac measure. But in QM, you can always multiply a state by a complex number and it still describe the same physical situation. So the distribution would really have to be a something like an indicator function. I'm not sure if this can be done. Maybe if you modify the theory a little and switch to the projective space. Then \lambda isn't the wave function itself, but rather an equivalence class of wave functions.

However, i have to admit that i misunderstood this at first. I thought you were considering arbitrary distributions of the \lambda's, as one usually does it in the derivation of Bell's theorem and this looked like a hopeless task. If you restrict the distributions you consider to only special cases, it looks much more feasible.

Bohr's Copenhagen interpretation insisted that the directly perceivable macroscopic classical world existed. (He literally insisted repeatedly on this, in the context of saying that all empirical data was ultimately statements about such macroscopic things.) So strictly speaking, the Copenhagen interpretation involves dividing the world into two realms -- the classical/macro realm and the quantum/micro realm. The former just unproblematically exists, essentially by postulate, and so there are lots and lots of beables there.

Well, i think that the term "Copenhagen interpretation" is used more loosely today. I would probably consider myself a quantum instrumentalist. I don't assume that the classical world exists (that means I'm agnostic but rather tend to neglect its existence if problems emerge). In fact, the measurement apparatus itself and everything else should also behave quantum mechanically. We just don't include it in our models most of the time (but we could do it and it leads to very useful results, see decoherence). The quantum-classical split doesn't have an ontological status in my opinion. We use classical theories only to interpret the results of measurements. They aren't part of the quantum theory itself. That means that if we obtain a value for a position measurement of a quantum particle for example then if we were to use a classical theory for the further description of the system (instead of quantum mechanics), then it would probably be best to assign the obtained value to the position variable of that classical theory in order to model the situation best. That doesn't mean that the quantum particle has suddenly acquired a position. It just means that we have mentally assigned a classical position to it in order to get a more intuitive understanding of the situation. We do this for the sole reason that we have more intuition for classical theories than for quantum theories. If this view of quantum theory makes me a non-Copenhagenist, so be it, but i think it is shared by most physicists at least in a similar way. I don't persist on being an advocate of any particular interpretation.

So, long story short, classical theories aren't part of the quantum description. They are only an interpretational tool. Quantum mechanics doesn't assign definite values to observables. It assigns only probability distributions, from which we can calculate expectation values. Nothing more. Quantum mechanics doesn't tell us that the outcome of a position measurement will be "5". It just tells us that if we prepare the system identically and perform the same measurement 100 times, then if we calculate the mean value, we will get "5". In fact, QM doesn't even have a mechanism to predict individual outcomes of an experiment. It's not a theory like classical mechanics where you just don't know the exact positions and momenta and thus supplement it with a probability distribution. In fact, QM is solely probability. It's not a theory about an underlying reality. Individual outcomes aren't even observables of the pure quantum theory. So how can they be beables of the theory? You said yourself that a beable is

"whatever a certain candidate theory says one should take seriously, as corresponding to something that is physically real."

But quantum mechanics doesn't say that one should take the individual outcomes seriously, because it's not a theory of individual outcomes. It doesn't predict individual outcomes. They aren't an element of the theory (unless you artificially add them like in Bohmian mechanics, but i only talk about standard QM here). Individual outcomes are external things that aren't part of the theory. And if they aren't part of the theory, they can't be beables of the theory. In your own paper, you quoted Bell saying that beables are always to be viewed with respect to a particular theory (in our case QM). They aren't global things that apply to all theories. I think that this is even the most relevant difference between Bohmian mechanics and standard QM. Assinging beable status to individual outcomes would probably cast standard QM almost into being Bohmian mechanics.

In short, the "measurement outcomes" most certainly *are* beables for Copenhagen. Bohr himself insisted on it specifically. And to deny such outcomes beable status -- for *any* theory -- is frankly borderline crazy. We are talking here about concrete things like which way a certain pointer in a certain lab pointed at a certain time. To deny "physically real" status to such things is ... well ... to approach solipsism.

As i said, you could also include the measurement apparatus and thus the pointer into the quantum description like the decoherence people do. Then decoherence tells you that the quantum state of the pointer will be sharply peaked over certain values after a very short time and the peak is getting sharper and sharper every nanosecond, but in fact it will never reach an exact eigenstate, so technically it's always in a superposition unless you wait an infinitely long amount of time, even though the peak will become so sharp that it practically makes no sense to talk about superpositions anymore. In that sense, the pointer of the measurement apparatus -- if described quantum mechanically -- behaves no differently than a quantum particle. We can compute only probability distributions. It's just that macroscopic objects have sharply peaked quantum states, just like particles shortly after their measurements. Sharply peaked quantum states are the classical limit of quantum theory, so to speak, but they aren't classical. They are only classical enough in the sense that the corresponding classical theory would provide a good approximation to the quantum description. I really don't have a problem with that. Especially i don't see why this would approach solipsism. I really have spent a considerable amount of time thinking about this kind of stuff. I haven't always thought about it this way.

You have to view it this way: A physical model is to nature like the word "banana" is to the yellow thing that you can buy in the supermarket. Ceci n'est pas une pipe (google it if you don't recognize it). Theories only describe our world. Some theories have just turned out to be useful. It's the theories that we classify by words like "local", "realistic" and others. It's not nature itself. If i say that standard QM doesn't have a beable corresponding to individual outcomes, this means that standard QM isn't concerned with invidivual outcomes. It doesn't make predictions about them. It only describes some aspects of the world, just like Newtonian gravity doesn't describe nuclear physics. Still, we can classify these theories using words like "local". You wouldn't say that Newtonian gravity can't be classified as "local" or "nonlocal" just because it has no means to describe nuclear physics. QM has no means to describe individual outcomes. Maybe that doesn't satisfy you, but it's enough for virtually every application i can think of and it doesn't prevent it from being classified. Maybe there is a deeper theory that can talk about individual outcomes and has them as beables. But yet, there is only Bohmian mechanics and i don't think that it has any particular advantage over standard QM.

What exactly you say is going on physically at the micro-level of course depends on which theory you're talking about and in particular what objects have beable status for the theory in question. But I fundamentally disagree about the last part of what you write. An individual measurement absolutely does tell us something about nature. Think about what an individual measurement means, concretely and physically. It means (at least) that some macroscopic object (think: pointer) moved a certain way. That is something we can directly perceive. It is pre-eminently physical, a fact about nature. (This was also one of the points that I guess you glossed over in the "long, vague paragraphs of text".)

But an individual measurement tells us very little about nature. It could as well be a measurement error. With only one datapoint, we are completely unable to tell. The actual value of the measurement is almost useless. We need a larger dataset to gain real information. The standard deviation is equally important as the measurements themselves. Measurements are always imprefect and physics somehow has to deal with this imperfection. There is no way out of this. There will never be a perfect measurement apparatus and thus, physics can't possibly live without statistics. This is a fundamental fact that can't be overcome. The only interesting values about a measured dataset are it's statistical properties. If you measure a single datapoint, say the position of an atom, to be "5", then this just tells you that the position of the atom might have been "5", but it might as well not have been "5", because the apparatus just gave you a wrong number due to the intrinsic imperfection of physical measurements. Even worse, if you measure the value "5", then this value is almost certainly wrong, because a measuremt error of "0" would be infinitely unlikely. We can never reliably reproduce individual outcomes, but we can reproduce their statistics. That's by the way also one of the main reasons why I'm willing to give up the beable status of individual measurements so easily.

Some do, some don't. But (as discussed above) this doesn't matter. You can give it either status you want, and "Copenhagen QM" is still nonlocal.

But maybe giving up the individual outcomes as beables makes it local. In fact, i could imagine that this would make Bell's locality definition equivalent to the definition that quantum field theorists use, which would be really cool in my opinion.

 

[rlduncan]

In this case, a "hidden variable" λ is just a specification of who gets the green ball. So there are two possible values of λ: Ag and Bg. So

P(Alice-green, Bob-red | λ=Ag) = 1
P(Alice-green, Bob-red | λ=Bg) = 0
P(Alice-red, Bob-red | λ=Ag) = 0
P(Alice-red, Bob-green| λ=Bg) = 1

I may be wrong, but this does not seem correct. There are only two outcomes for this case:
1) Alice-green, Bob-red
2) Alice-red, Bob-green.

So let A=Alice gets green and B=Bob gets red. From the generalized conditional probability rule:

P(A|B) = P(A)*P(B|A) where P(A) = 0.5 and P(B|A) = 1 (if Alice got green, then it’s a 100% certainty Bob got red). So

P(A|B) = P(A)*P(B|A) = (0.5)(1) = 0.5 Eq(1)

which is the correct result for this special case. (The same may be written for A=Alice gets red and B=Bob gets green.)

Now according to Bell’s logic:

P(AB|λ) = P(A|λ) * P(B|λ) Eq(2)

If A and B are independent events then P(B|A) =P(B) and Eq(1) reduces to
P(A|B) = P(A)*P(B) = (0.5)(0.5) = 0.25 which is incorrect and Bell certainly understood this.

So let me rephrase. What is λ in Eq.(2)? What are the values for the terms P(A|λ) and P(B|λ)? I am assuming that P(AB|λ) = 0.5 the same answer calculated in Eq.(1). I can’t reason logically that Eq(1) and Eq(2) are equivalent. Any clarification would be appreciated.

P.S. I pose these questions in part to understand the challenge by ttn to explain how to account for the perfect correlations when the analyzers point to the same angle.

 

[ttn]

I may be wrong, but this does not seem correct. There are only two outcomes for this case:
1) Alice-green, Bob-red
2) Alice-red, Bob-green.

So let A=Alice gets green and B=Bob gets red. From the generalized conditional probability rule:

P(A|B) = P(A)*P(B|A) where P(A) = 0.5 and P(B|A) = 1 (if Alice got green, then it’s a 100% certainty Bob got red). So

P(A|B) = P(A)*P(B|A) = (0.5)(1) = 0.5 Eq(1)

which is the correct result for this special case. (The same may be written for A=Alice gets red and B=Bob gets green.)

Now according to Bell’s logic:

P(AB|λ) = P(A|λ) * P(B|λ) Eq(2)

If A and B are independent events then P(B|A) =P(B) and Eq(1) reduces to
P(A|B) = P(A)*P(B) = (0.5)(0.5) = 0.25 which is incorrect and Bell certainly understood this.

So let me rephrase. What is λ in Eq.(2)? What are the values for the terms P(A|λ) and P(B|λ)? I am assuming that P(AB|λ) = 0.5 the same answer calculated in Eq.(1). I can’t reason logically that Eq(1) and Eq(2) are equivalent. Any clarification would be appreciated.

P.S. I pose these questions in part to understand the challenge by ttn to explain how to account for the perfect correlations when the analyzers point to the same angle.

Actually what stevendaryl wrote was exactly right. Once you conditionalize on λ (a *complete* description of the state of the balls prior to measurement) all the probabilities are either 0 or 1 since there is no fundamental randomness here according to the theory in question (which is "common sense" or "classical physics" or some such).

The relation to the case of quantum particles (my challenge) is as follows. Suppose you say that you have a different theory, namely, that neither ball has any definite color while they're still in the boxes. They only acquire a definite color through some random process when looked at. So now λ does *not* include the real pre-observations colors of the balls because there is (according to this alternative theory) no such thing. But now, if the model is *local* -- i.e., if each ball (when observed) switches to red or green with 50/50 probability independently of anything happening far away -- then the theory will predict that 25% of the time the balls are both red, etc.

So if there is some experimental data showing that, actually, the balls are always different colors, we face a choice. Either they had those colors all along and the colors are simply revealed to us when we look (hidden variables!), or the colors are indefinite until observation happens but there is some nonlocality in the way observation makes the colors pop into existence (e.g., Alice's observations randomly makes her ball become either red or green *and makes Bob's distant ball become the opposite color*).

This is exactly what you should be thinking about to appreciate why locality --> "realism", i.e., why the *only* way to explain the perfect correlations *locally* is to posit "hidden variables".

 

[ttn]

...

If I understand you correctly, you are saying that QM cannot account for the fact that something like a pointer (or a table or a cat or a planet) exists with definite properties.

You're getting lost in questions like whether/how the "cut" can be pushed around so that macroscopic stuff is described on the wave function side, whether QM *predicts* exactly what the outcome of an experiment will be, whether we can become omniscient about the state of some micro-thingy from only a single measurement on it, etc.

But none of that is relevant to the main point here. We don't need QM or any other fancy theory to tell us that pointers point in particular directions, that there's a table in front of me, a cat on the bed, etc. Physical facts like that are just available to direct sense perception. We know them more directly, with more certainty, than we can possibly ever know anything about obscure microscopic things. Now here is the simple plain fact. To whatever extent you are right that QM cannot account for these sorts of facts (and personally I think you are not right at all, i.e., I think Copenhagen QM *does* account for them, and it was one of Bohr's few valid insights to recognize that it is *crucial* that it be able to account for them) it ceases to be an empirically adequate theory.

 

[stevendaryl]

I may be wrong, but this does not seem correct. There are only two outcomes for this case:
1) Alice-green, Bob-red
2) Alice-red, Bob-green.

In English, we explain this case as follows: (Let me change it slightly from previously)

There are two boxes, one is labeled "Alice", to be sent to Alice, and the other labeled "Bob" to be sent to Bob. We flip a coin, and if it is heads, we put the green ball in Alice's box, and the red ball in Bob's box. If it is tails, we put the red ball in Alice's box, and the green ball in Bob's box.

In this case, the hidden variable \lambda has two possible values, H, for "heads" and T for "tails". Then our probabilities are
(letting A mean "Alice gets green" and B mean "Bob gets red".)

• P(H) = P(T) = \frac{1}{2}
• P(H T) = 0
• P(A \vert H) = P(B \vert H) = 1
• P(A \vert T) = P(B \vert T) = 0

We can compute other probabilities as follows:

• P(AB) = P(AB \vert H) \cdot P(H) + P(AB \vert T) \cdot P(T) = \frac{1}{2}
• P(A) = P(A \vert H) \cdot P(H) + P(A \vert T) \cdot P(T) = \frac{1}{2}
• P(B) = P(B \vert H) \cdot P(H) + P(B \vert T) \cdot P(T) = \frac{1}{2}
• P(A \vert B) = P(AB)/P(B) = 1

Bell's criterion for the case of A and B being causally separated is not

P(A \vert B) = P(A)

(which is false). Instead, it's

P(A \vert B \lambda) = P(A \vert \lambda)
where \lambda is a complete specification of the relevant information in the common past of A and B, which is true.