An abstract long-distance correlation experiment

[ddd123]

... ruled out only under the assumption of a local hidden variable theory with signals moving independently along the rays to Alice and Bob. But this assumption is too strong to have implications when the signal is a field rather than particles.

Suppose the detectors are at the sides of the source, all on the same axis. Whatever the source is emitting makes the detectors show EPR correlations with light-speed timing. The field would have to instantly jump double the distance to "tell the other side" what choice of measurement was made so to make the correlations show up.

 

[A. Neumaier]

Suppose the detectors are at the sides of the source, all on the same axis. Whatever the source is emitting makes the detectors show EPR correlations with light-speed timing. The field would have to instantly jump double the distance to "tell the other side" what choice of measurement was made so to make the correlations show up.

I know, but your observation is independent of what I was asserting. The speed of light is nowhere used in the description or analysis of the experiment, except to conclude ''nonlocality''.

 

[ddd123]

A nonlocal field reminds me of pilot wave. It's not less weird.

 

[maline]

For I think the same concerns that you raise above apply to the single-photon nonlocality experiment, although the latter has a fully classical field explanation.

The classical field explanation is precisely of the local "hidden" variable type. The light has a particular polarization at every point, which determines its behavior. There is no reason to assume that a signal hitting a beam splitter goes in only one direction, other than the weird quantum fact that there exist "single photons" that get fully absorbed at a single point!

 

[A. Neumaier]

The classical field explanation is precisely of the local hidden variable type. The light has a particular polarization at every point, which determines its behavior. There is no reason to assume that a signal hitting a beam splitter goes in only one direction, other than the weird quantum fact that there exist "single photons" that can only be detected at one point at most!

Did you read my slides?

I gave a local hidden variable argument of precisely the kind that was used by Bell and found a Bell-type inequality that was violated by the prediction of quantum mechanics. According to your criticism, there should be a fault in my formal reasoning since repeating the analysis using instead the Maxwell equations gives full agreement with the quantum predictions.

So please point out where my arguments are faulty instead of arguing in a roundabout way that is too vague to spot the problems! it was Bell's accomplishment to do this for the EPR problem and thus turn it from a philosophical issue into something that can be investigated in a scientific manner. So please argue on the level of equations rather than philosophy if you want to make a scientific point in the spirit of Bell!

 

[maline]

In your "inequality derivation" you assumed that the photon takes one path at the splitter. Bell's argument does not rely on any such assumption.

 

[A. Neumaier]

In your "inequality derivation" you assumed that the photon takes one path at the splitter. Bell's argument does not rely on any such assumption.

Isn't it already extremely weird to allow that a classical local hidden variable photon travels along several beams? I don't think that it is satisfying to explain away weirdness by basing the explanations on weird assumptions.

 

[zonde]

This is apparent from a (simpler) single-photon nonlocality experiments such as that discussed in my slides here (slides 46-59). The argument there doesn't extend to the setting under discussion here but shows that the assumptions of Bell are tied to an implicit particle assumption.

Here is field type explanation of Bell inequality violation http://arxiv.org/abs/0906.1539. But it needs to exploit loophole to do that.

 

[maline]

Isn't it already extremely weird to allow that a classical local hidden variable photon travels along several beams?

Classically, there are no photons, only an EM wave. Of course it spreads through space along all possible paths. Nevertheless, its polarization is a local variable, because it propagates at light speed.

 

[A. Neumaier]

an EM wave. Of course it spreads through space along all possible paths

No. There are no paths in a field context. And as any experimenter in classical optics knows, if you input a polarized electromagnetic wave focussed in a beam (in the paraxial approximation) into a beam splitter, the output will be a polarized electromagnetic wave focussed just along two beams. And its polarization is bilocal, not local. This is why one gets the quantum mechanical results and not the local hidden variable results.

 

[maline]

There are no paths in a field context

"Along all possible paths" was just a way of saying "as per Maxwell's equations". The point is that classically the wave must travel both ways, and there is nothing weird about that. "Photon" is purely a quantum- or rather a QFT- concept (and to me is indeed quite weird).

And its polarization is bilocal, not local. This is why one gets the quantum mechanical results and not the local hidden variable results

The word "local" in "local hidden variable" does not mean "localized to a particular region". It means "respecting the principle of locality"- the variable can be described as a function on spacetime (including delta functions) and values at particular points depend only on the past light cone of those points. Classical polarization of a wave definitely qualifies.

 

[wle]

This can become negative (uniformly random output independent of input), hence is not yet good. What about $W=\max(0,\min(6\bar F-2\bar E-4,1))$? This would satisfy my requirements, and makes the specific case you described completely weird while leaving local hidden variable results not weird at all.

If this fits with your requirements, sure. My opinion isn't really important here. Other participants in this thread have indicated that they want a measure of nonlocality and are apparently happy if they have one that's a function of only your matrices $E$ and $F$, so I posted one. The only reasons I used the normalisation I did are a) I know that quantum physics can attain $- \bar{E} + 3 \bar{F} = 2.5$, but I don't have a proof that this is the maximum that is consistent with QM, and b) it's easy to define a set of hypothetical conditional probabilities (almost certainly not allowed by QM) that attain the algebraic limit $- \bar{E} + 3 \bar{F} = 4$ without allowing instantaneous signalling (in the sense that the marginal statistics on Alice's side are independent of Bob's measurement choice and vice versa).

 

[A. Neumaier]

The word "local" in "local hidden variable" does not mean "localized to a particular region". It means "respecting the principle of locality"- the variable can be described as a function on spacetime (including delta functions) and values at particular points depend only on the past light cone of those points.

Local in local hidden variable theories cannot mean anything related to relativity theory - all of quantum mechanics is purely nonrelativistic!

Indeed, I have never seen a Bell-type argument where formal use was made of the the fact that values depend or do not depend on the past light cone. The arguments never involve space or time at all, only simultaneity, which is intrinsically nonrelativistic!

 

[A. Neumaier]

a) I know that quantum physics can attain $- \bar{E} + 3 \bar{F} = 2.5$, but I don't have a proof that this is the maximum that is consistent with QM, and b) it's easy to define a set of hypothetical conditional probabilities (almost certainly not allowed by QM) that attain the algebraic limit [...] 4

Anything with $- \bar{E} + 3 \bar{F} \ge 2.5$ has degree of weirdness 1 according to my amended formula.

 

[wle]

Thank you for the post. Can you add a sample calculation with 2 matrices to help some of us that may be a little confused by the terminology. Regards.

I'll elaborate for the quantum example. The example I had in mind is where Alice and Bob each have a spin 1/2 particle entangled in the state $\lvert \Psi^{-} \rangle = \bigl( \lvert 0 \rangle_{\mathrm{A}} \lvert 1 \rangle_{\mathrm{B}} - \lvert 1 \rangle_{\mathrm{A}} \lvert 0 \rangle_{\mathrm{B}} \bigr) / \sqrt{2}$ and can measure the spin projections respectively along the angles $\alpha_{x}$ and $\beta_{y}$, $x, y \in \{0, 1, 2\}$ on the $\sigma_{z}-\sigma_{x}$ plane. The well known prediction by QM is that, depending on the angles, Alice and Bob get the results (noted '0' and '1') with joint conditional probabilities $$\begin{eqnarray*}
P(00 \mid xy) &=& P(11 \mid xy) &=& \frac{1 - \cos(\alpha_{x} - \beta_{y})}{4} \,, \\
P(01 \mid xy) &=& P(10 \mid xy) &=& \frac{1 + \cos(\alpha_{x} - \beta_{y})}{4} \,.
\end{eqnarray*}$$ The result I described is obtained with the choices of angles (in degrees) $$\begin{eqnarray*}
\alpha_{0} &=& \beta_{0} &=& 0^{\circ} \,, \\
\alpha_{1} &=& \beta_{1} &=& 120^{\circ} \,, \\
\alpha_{2} &=& \beta_{2} &=& 240^{\circ} \,,
\end{eqnarray*}$$ so that the angular difference $\alpha_{x} - \beta_{y}$ is always $0^{\circ}$, $\pm 120^{\circ}$, or $\pm 240^{\circ}$. Basic trigonometry says that $\cos(0^{\circ}) = 0$ and $\cos(\pm 120^{\circ}) = \cos(\pm 240^{\circ}) = -1/2$, so that $$\cos(\alpha_{x} - \beta_{y}) = \begin{cases}
0 &\text{if } x = y \\
-1/2 &\text{if } x \neq y
\end{cases} \,.$$
Inserted into the probabilities above, this gives $$\begin{eqnarray*}
P(00 \mid xy) &=& P(11 \mid xy) &=& 0 \,, \\
P(01 \mid xy) &=& P(10 \mid xy) &=& 1/2
\end{eqnarray*}$$ if $x = y$ and $$\begin{eqnarray*}
P(00 \mid xy) &=& P(11 \mid xy) &=& 3/8 \,, \\
P(01 \mid xy) &=& P(10 \mid xy) &=& 1/8
\end{eqnarray*}$$ if $x \neq y$. In other words, Alice and Bob get perfectly anticorrelated results if they choose the same angle settings ($x = y$) and partially correlated results if they choose different angle settings ($x \neq y$). The elements $E_{ab}$ and $F_{ab}$, $a, b \in \{0, 1\}$, of the matrices $E$ and $F$ are just the probabilities $P(ab \mid xy)$ averaged over the cases $x = y$ and $x \neq y$, i.e., $$\begin{eqnarray*}
E_{ab} &=& \frac{1}{3} \sum_{x} P(ab \mid xx) \,, \\
F_{ab} &=& \frac{1}{6} \sum_{x \neq y} P(ab \mid xy) \,.
\end{eqnarray*}$$ Here each of the contributing probabilities is the same in both cases, so the matrices are just $$E = \begin{bmatrix} 0 & 1/2 \\ 1/2 & 0 \end{bmatrix}$$ and $$F = \begin{bmatrix} 3/4 & 1/4 \\ 1/4 & 3/4 \end{bmatrix} \,.$$ Finally, using the definition from my earlier post, this gives $\bar{E} = 0 - 1/2 - 1/2 + 0 = -1$ and $\bar{F} = 3/8 - 1/8 - 1/8 + 3/8 = 1/2$, which produces $- \bar{E} + 3 \bar{F} = 2.5$.

Does this help? I didn't know which part of the notation you weren't following so I aimed to explain the example thoroughly.

 

[wle]

So please point out where my arguments are faulty instead of arguing in a roundabout way that is too vague to spot the problems!

The experiment depicted on p. 47 of your slides doesn't fit the format of a Bell experiment. In particular, you have only one detector and it is possible for it to be influenced by both filters $F(B_{1})$ and $F(B_{2})$ without any faster-than-light communication. This -- and this is a big difference from Bell -- makes it impossible to derive constraints on what the detector in your setup can register based on relativistic causality alone. In Bell's theorem the point was to derive constraints on possible correlations assuming relativistic causality and very little else. This is why there are at least two spatially separated detectors in a Bell experiment and why there's a lot of importance put on having the measurements performed and results recorded nearly simultaneously: those are minimum conditions necessary in order for slower-than-light causation to actually become a constraint on anything.

Indeed, I have never seen a Bell-type argument where formal use was made of the the fact that values depend or do not depend on the past light cone.

Here's some by Bell himself:

1. J. S. Bell, "The theory of local beables", CERN-TH-2053 (1975).
2. J. S. Bell, "Bertlmann's socks and the nature of reality", CERN-TH-2926 (1981).
3. J. S. Bell, "La nouvelle cuisine", doi:10.1017/CBO9780511815676.026 (1990).

Of these, 1. and 3. are very explicitly grounded in relativistic causality; in both, Minkowski diagrams depicting light cones are used as aides to the argument. 2. is also worth a read. It isn't so explicit about the role of relativity but the presentation is still kept very generic and Bell emphasises that the argument does not depend on, say, some specific model of "particles".

I linked to scans of 1. and 2. that are freely available online, though it's probably possible to find nicer reprints elsewhere. The link for 3. is unfortunately a paywall. All three are included in the second edition of "Speakable and Unspeakable in Quantum Mechanics", printed in 2004.

Some more recently written introductions which draw heavily on Bell's writings, including the above three essays:

1. T. Norsen, "John S. Bell’s concept of local causality", Am. J. Phys. 79, 1261 (2011), arXiv:0707.0401 [quant-ph].
2. S. Goldstein et. al., "Bell's theorem", Scholarpedia 6, 8378 (2011).

 

[A. Neumaier]

in both, Minkowski diagrams depicting light cones are used as aides to the argument.

I know that Bell (like earlier Einstein) used causality to motivate the experiment and to deduce nonlocality, but my emphasis was on ''formal use made of'' it. No formula involves anything relativistic - only the talk around it does. But the relevant physics is always in the formulas only, the accompanying talk is only interpretation. That's why multiple interpretations abound, while the formalism is universally agreed upon.

I had simplified the setting since the nonlocality part is still present and invites essentially the same weirdness considerations: There is no intuitively natural way of explaining how the detector can respond as it does - except for having the universe conspire to collect in a mysterious way the nonlocal information and turn it into the appropriate statistical output.

From my point of view, nothing more mysterious happens in Bell's experiment, since exactly the same algebra is used, and the speed of light argument is extreaneous to the whole formal setting. The quantum mechanics is nonrelativistic since simultaneity is essential, nowhere in the algebra the unity of space and time characteristic for relativity appear, and nowhere the Lorentz group figures.

 

[zonde]

I gave a local hidden variable argument of precisely the kind that was used by Bell and found a Bell-type inequality that was violated by the prediction of quantum mechanics. According to your criticism, there should be a fault in my formal reasoning since repeating the analysis using instead the Maxwell equations gives full agreement with the quantum predictions.

Isn't it already extremely weird to allow that a classical local hidden variable photon travels along several beams? I don't think that it is satisfying to explain away weirdness by basing the explanations on weird assumptions.

You didn't take Bell assumptions. If you modify Bell assumptions because you think they are weird does not change the fact that the assumptions used by you are not the ones used by Bell.

 

[A. Neumaier]

Let us return to the main topic.

Concerning Stage 2 (post #49), we now have one qualifying degree of
weirdness (post #112, fully justified in post #115) and an expression
of dissatisfaction about trying to quantify weirdness at all (post #98).

Leaving Stage 2 still open for a while, I'll begin with the next stage,
where I promised to state my own interpretation of the experiment,
and why I think the results are not weirder than what one finds
classically in other situations. My interpretation will extend over
two stages: Stage 3, where I make some general remarks that are
independent of what is discussed in Stage 2, and (when Stage 2
is completed) Stage 4, where I use the results of Stages 2 and 3 to
complete my view on the matter.

(Note: Stages 2 and 3 are now also closed for discussion.
Stage 4 begins in post #187.)

Stage 3 is opened with the following observations, whose discussion
is invited. My observations at this stage are completed in post #174,
with a discussion of how weirdness and knowledge are related.

In our setting, assume for the moment that the nature of Norbert's
signals are known to everyone, and are of the kind consistent with
quantum mechanics but inconsistent with Bell-type assumptions.
Assume also that there is a human Alice behind the dumb machine Alice.

Under these conditions I want to discuss what the human Alice
knows about Bob's results after she has completed her experiments.

My claim is that she knows nothing definite at all.

For the results Bob gets depend on what he is doing, and she is not
informed about the latter. At best she can draw conditional inferences
''If Bob's pointer position was set to ... then his results were ...''.
This is closer to guesswork of the form we use in medical diagnostics
when decisive facts are absent than to scientific knowledge of the kind
we can find in standard textbooks, and to engineering knowledge encoded
in properly working machines.

The knowledge that Alice has feels more like what we know about an
(ideal) pendulum when its initial conditions are unknown - we know the
general structure of the possible configurations, but we don't know
anything about the configuation itself. If we take the analogy seriously
we conclude that [given Norbert's fixed signalling strategy]
Nature solves an initial-value problem with two inputs
(pointer settings) and two outputs (color of response) - that on Alice's
side and that on Bob's side. The joint output depends on both inputs.

This dependence is Bell's form of nonlocality - demonstrated by this
kind of experiments and quite obvious from this way of thinking about
it, even without an experimental proof by the violation of corresponding
Bell inequalities.

Remarkably, Bell's findings wouldn't have seemed weird at the end of the
19th century - classical field equations such as the heat equation also show
this kind of nonlocality!
Nonlocality is classically intrinsic even to Newtonian mechanics in its
original form where celestial bodies act instantaneously over
arbitraily large distances. It is a standard part of nonrelativistic
classical mechanics. So why should nonlocality count as weird?

Being already manifestly present in nonrelativistic multiparticle
classical mechanics, it is no surprise at all that it is also present in
nonrelativistic multiparticle quantum mechanics such as Bell-type
experiments! Note that essentially all analysis of Bell nonlocality is
done in a nonrelativistic framework! Plus lip service paid to relativity,
in a form that doesn't enter at all into the formulas...

To impose weirdness by invoking arguments involving the speed of light
in an otherwise nonrelativistic framework also makes the heat equation
seem weird since a change in temperature at one place immediately
affects the temperature everywhere else.
... and Newton's celestial mechanics since the change in position of one
celestial body immediately affect the positions everywhere else.

The quibbles with this form of nonlocality are caused by a superficial
understanding of relativity theory and the use of superficial relativity
arguments in an explictly nonrelativistic classical or quantum setting.
What seems to be unnatural or weird is solely due to mixing two
incompatible settings.

If one attempts to disentangle the two settings interesting things happen:

On the purely nonrelativistic level, all weirdness has disappeared;
things are no worse in quantum mechanics than in classical celestial
mechanics or fluid mechanics.

On the other hand, one can try to see what happens when one looks
at classical relativistic multiparticle theories. Once one starts looking for
these (I challenge you to do such a search yourself) one finds that from
the outset, they are plagued with tremendous weirdness!

Clearly, it is the particle picture that - classically! - introduces this
weirdness into relativity theory since classical relativistic field theories
have no problem at all as long as one doesn't introduce point particles
into them. It thus appears that in quantum mechanics of point particles
the classical weirdness is even softened since it appears only in situations
that take a lot of effort to prepare, and disappears completely once one
consistently stays in the realm of quantum field theory.

 

[stevendaryl]

... ruled out only under the assumption of a local hidden variable theory with signals moving independently along the rays to Alice and Bob. But this assumption is too strong to have implications when the signal is a field rather than particles.

The sense of "local" that is important to Bell's analysis is the issue of whether something taking place at Alice's detector has an effect on Bob, or vice-versa.

So let me grant the possibility that the entire environment--the whole rest of the universe--works together in a nonlocal way to establish the two outcomes at Alice and Bob. Then the question becomes whether it is possible for Alice and Bob to make up their minds at the last minute as to which detector setting to choose. This is sometimes called the "free will assumption", but it doesn't actually need to rely on anything mystical about consciousness. It's just that Alice and Bob can base their choice on absolutely anything, such as a radioactive decay of a uranium atom, or some characteristic of the light from a distant star, etc.

Let's split up the universe into three parts:

1. The part \lambda relevant to the production of the twin-pair.
2. The part \alpha relevant to Alice's choice of her detector setting.
3. The part \beta relevant to Bob's choice of his detector setting.

So, if you are claiming that the details of the whole universe make the outcomes deterministic, then that would seem to me to imply to me that there is a pair of functions determining the outcomes:

• F_{Alice}(\alpha,\beta,\lambda) = the probability of Alice getting spin-up, given \alpha, \beta, \lambda
• F_{Bob}(\alpha,\beta,\lambda) = the probability of Bob getting spin-up, given \alpha, \beta, \lambda

The key question relevant to Bell's argument is really about whether the three parameters \alpha, \beta and \lambda can be varied independently. If they cannot be, that's pretty weird. If they can be, then Bell's analysis goes through, showing that for any such functions F_{Alice} and F_{Bob}, Alice's result must depend, in a FTL way, on conditions at Bob, or Bob's result must depend, in an FTL way, on conditions at Alice.

 

[A. Neumaier]

Let's split up the universe into three parts:

1. The part λ relevant to the production of the twin-pair.
2. The part α relevant to Alice's choice of her detector setting.
3. The part β relevant to Bob's choice of his detector setting.

Why isn't there a fourth part $\gamma$, relevant to both Alice's and Bob's choice of detector setting? You make the assumption that this part is empty, but I cannot see a good reason for it. Thus $F_A$ and $F_B$ also depend on $\gamma$, and Bell's argument breaks down.

 

[georgir]

I am confused... so you think the weirdness comes from thinking in terms of particles...
"classical relativistic field theories have no problem at all as long as one doesn't introduce point particles"
But do you say changes in the fields that you have in that case still propagate at a finite speed (you used the term "relativistic", so I guess yes)? Then such a model can not reproduce Bell violations.
On the other hand if you have instantaneous or FTL changes in the fields, you already have non-locality, and hence weirdness.

I do not see how the FTL field changes in Newtonian mechanics (which is now known to be wrong) not being considered weird in the past supports your view that FTL changes in QM should not seem weird now.

 

[A. Neumaier]

you think the weirdness comes from thinking in terms of particles...

Yes, most of it. All of it comes from mixing in an inappropriate way different intuitions coming from incompatible formal settings.

Note that I discuss classical weirdness by analogy rather than by giving a model that would explain the quantum results. Quantum mechanics makes predictions different from classical mechnaics hence shouldn't be explained in terms of the latter. I mainly argued two points:

1. Introducing no faster than light arguments into an otherwise completely nonrelativistic setting produces contradictions already in classical theories. This is relevant - even if though classical theories are known to be approximations only - since weirdness is clearly primarily deviation from classical intuition. Hence if classical thinking in approximate classical theories such as celestial mechanics or hydromechnaics is already incompatible with no faster than light arguments, the weirdness is already due to this and not primarily to the quantum features.

2. In a classical relativistic setting, the notion of a 2-particle system is already ill-defined and fraught with conceptual difficulties. Only a single particle has a good relativistic description.

 

[zonde]

Why isn't there a fourth part $\gamma$, relevant to both Alice's and Bob's choice of detector setting? You make the assumption that this part is empty, but I cannot see a good reason for it. Thus $F_A$ and $F_B$ also depend on $\gamma$, and Bell's argument breaks down.

Why $\gamma$ can't be included into $\lambda$ ? What's so specific about $\gamma$ ?

 

[ddd123]

Neumaier, since back when instantaneous field were the norm we found out that spacetime is locally Minkowskian (as verified by atomic clocks on fast airplanes), so whatever the framework of the theory is, the Bell violations as proven by experimental loophole-free Bell tests (especially those avoiding the communication loophole) are still weird even if you abandon the particle idea.

 

[Mentz114]

Neumaier, since back when instantaneous field were the norm we found out that spacetime is locally Minkowskian (as verified by atomic clocks on fast airplanes), so whatever the framework of the theory is, the Bell violations as proven by experimental loophole-free Bell tests (especially those avoiding the communication loophole) are still weird even if you abandon the particle idea.

All the Minkowski/relativistic principles you are applying are verified for macroscopic scales. A bunch of 'events' are happening in only a few microseconds, and robots are telling you they happened in a certain order - and gave certain measurement results. We then ascribe other events as being simultaneous with those. I would not stake a penny on that assumption being right unless I saw it with my own eyes. Which is impossible.

Applying billiard-ball dynamics is not appropriate.

 

[ddd123]

So you would stake on the fact that, if we could maintain coherence for big enough distances, "nature's prank" would stop working? We've already gone a mile. I don't think there's any reason to believe that.

 

[A. Neumaier]

What's so specific about γ?

That (unlike $\lambda$, if I interpret its definition correctly) it is not determined by local information.

It is a nontrivial restriction to assume that every piece of information must have originated at a single local point. It is this restriction that excludes nonlocal correlations.

It is natural fur us humans to assume this since we can generate and transmit information only locally. But Nature (being far bigger than a human or a machine built by humans) is not necessarily bound in this way, and Bell-type experiments prove that it really isn't.

 

[A. Neumaier]

since back when instantaneous field were the norm we found out that spacetime is locally Minkowskian

So you say that weirdness is a function of our knowledge about Nature? Then after a century of having found out that Nature is quantum we should have long adapted our conception of weirdness to find the clash between classical relativistic thinking and quantum mechanics natural (non-weird) in the same way that we no longer find the clash between instantaneous action and relativity weird.

 

[maline]

Local in local hidden variable theories cannot mean anything related to relativity theory - all of quantum mechanics is purely nonrelativistic!

I don't see your point here. Bell's Theorem is a statement about which results cannot be reproduced by local, realist, counterfactual definite models. Such models, by definition, respect the light- speed boundary.

Indeed, I have never seen a Bell-type argument where formal use was made of the the fact that values depend or do not depend on the past light cone. The arguments never involve space or time at all, only simultaneity, which is intrinsically nonrelativistic!

If we allow FTL influences, then Bell's Theorem certainly does not apply! That's why we have models like Bohmian Mechanics or Continuous Reduction.

Here is a short sketch of Bell's logic:
1.Given locality, and spacelike separation, Alice's detector settings and measurement result have no effect on Bob's measurement result.
2.Therefore, Bob's results depend only on the signal in Bob's region, and his settings.
3.Given that, for any setting Bob chooses, there is a hypothetical scenario in which his result can be known before the measurement, the result must be fully determined by the signal in Bob's region, for any detector setting.
4. Therefore, the only way probability enters is in the distribution of the signals: a probability of 3/4, say, for a measurement to find positive spin in some direction means that 3/4 of the signals are such that will definitely give the positive result for that measurement.
5. Now comes the formal algebraic part: there is no distribution that matches the quantum (experimental) probabilities for all settings.
6.Conclusion: one of the assumptions - locality, realism, or counterfactual definiteness- is not true of Nature.

Without the "no FTL" assumption, the measurements can affect each other, and the argument does not begin.

 

[maline]

This is relevant - even if though classical theories are known to be approximations only - since weirdness is clearly primarily deviation from classical intuition. Hence if classical thinking in approximate classical theories such as celestial mechanics or hydromechnaics is already incompatible with no faster than light arguments, the weirdness is already due to this and not primarily to the quantum features.

The weirdness discussed here is not "deviation from intuition". It's more like "inability to form a picture of the fundamental reality". This is only relevant for models that are intended to be fundamentally accurate.

2. In a classical relativistic setting, the notion of a 2-particle system is already ill-defined and fraught with conceptual difficulties. Only a single particle has a good relativistic description.

I don't know of these difficulties. Please elaborate. Anyway, adding more problems does not a solution make!

Then after a century of having found out that Nature is quantum we should have long adapted our conception of weirdness to find the clash between classical relativistic thinking and quantum mechanics natural (non-weird)

We are trying. We have tried for a century, and thus far our efforts have been met with failure. This thread is part of the ongoing attempt.

 

[A. Neumaier]

Here is a short sketch of Bell's logic:
1.Given locality, and spacelike separation, Alice's detector settings and measurement result have no effect on Bob's measurement result.
2.Therefore, Bob's results depend only on the signal in Bob's region, and his settings.
[...]
6.Conclusion: one of the assumptions - locality, realism, or counterfactual definiteness- is not true of Nature.

The culprit is the form of locality assumed in 1. to be able to conclude 2. This form of locality is not realized in Nature. However, Assumption 1 is a far stronger assumption than what follows from relativity = Lorentz invariance alone.

This is what I mean when I say that the physical substance is always in the formulas and not in the story created around them, and that Bell's theorem isn't making the formal connection to relativity theory. The latter is obvious in your synopsis since the setting given is manifestly non-invariant and the Lorentz group isn't even mentioned.

Modern relativity is the claim that Nature is ruled by Lorentz invariant laws, nothing else. Every valid claim about conflicts with relativity must produce a contradiction with Lorentz invariance, not only with one of the fuzzy verbal phrases such as ''no FTL influences''. The (serious) step missing is to deduce from Lorentz invariance that, in a formally precise sense, there are ''no FTL influences'', and then to conclude Step 2 from this formally precise meaning of ''no FTL influences''.

 

[A. Neumaier]

The weirdness discussed here is not "deviation from intuition". It's more like "inability to form a picture of the fundamental reality".

To me it seems to articulate the ''inability to form a classical picture of the fundamental reality" - since the quantum picture is obviously an appropriate representation of fundamental reality. It allows us to predict and control a lot of stuff that 100 years ago were science fiction only.

 

[ddd123]

So you say that weirdness is a function of our knowledge about Nature? Then after a century of having found out that Nature is quantum we should have long adapted our conception of weirdness to find the clash between classical relativistic thinking and quantum mechanics natural (non-weird) in the same way that we no longer find the clash between instantaneous action and relativity weird.

The merit of these threads is that I've had my notion of weirdness clarified. At the very substance, it ended up being less about intuition and more about "undecidable" quandaries. Steveandaryl put it best imho: "what's weird is the lack of an answer to some basic questions about QM, particular to the EPR experiment". We can concoct some answers but they're all mutually exclusive, unfalsifiable and, worse, all have more or less an ad-hoc feel to them. They're not what physics' tradition considers elegant and sound; some are even on the wild speculations spectrum.

 

[A. Neumaier]

The merit of these threads is that I've had my notion of weirdness clarified.

Yes, that's the very purpose of the threads.

 

[maline]

However, Assumption 1 is a far stronger assumption than what follows from relativity = Lorentz invariance alone.

Yes, Bell locality is intended as a stronger assumption than "relativity holds". It is justified (for me) by:
1.The intuition that causation occurs from past to present to future, in an objective sense. Since relativity does not define regions outside the light-cone as "past" or "future", causation should be confined to this cone.
2.FTL signalling would imply a possibility of sending messages to the past, and I see no fundamental reason why signals should differ from other forms of influence.
Therefore, to me, the violation of locality is weird.

But the reason we got into "locality" was to explain the difference between EPR experiments and ones that can be explained classically, such as the polarization example in your slides. All relativistic classical descriptions are also local in Bell's sense. In those cases, "probability inequalities" indeed result only from assuming a particle concept,(which is "quantum" and certainly not classical). This is not the case for Bell's Theorem. No local realist model, whether involving particles, fields, or anything else, can violate the inequality.

Modern relativity is the claim that Nature is ruled by Lorentz invariant laws, nothing else.

That brings up another point: as far as I know, no Lorentz invariant description has ever been given for QM including measurements. To me this is a hint that something important is missing from the fundamentals.

the quantum picture is obviously an appropriate representation of fundamental reality. It allows us to predict and control a lot of stuff

Ability to predict and control does not imply understanding. By a "picture of reality" I mean the ability to answer simple questions like "if an electron propagates through space as a wave, and then is detected at one point, what happens to the rest of the wave?"

 

[A. Neumaier]

2. In a classical relativistic setting, the notion of a 2-particle system is already ill-defined and fraught with conceptual difficulties. Only a single particle has a good relativistic description.

I don't know of these difficulties. Please elaborate. Anyway, adding more problems does not a solution make!

This was not part of a solution - which is partly indicated in this post - but part of my argument that weirdness is not in the quantum part but in the particle part.

The problems involved in a classical multiparticle setting are addressed in a post of the PF thread ''Introduction to relativistic quantum mechanics and maybe QFT'' and the subsequent discussion,
together with the references provided there. Discussion of this point should be done there, not here.

By a "picture of reality" I mean the ability to answer simple questions like "if an electron propagates through space as a wave, and then is detected at one point, what happens to the rest of the wave?"

It would be enough to have a language that forbids asking questions such as this because they are meaningless.

 

[wle]

I know that Bell (like earlier Einstein) used causality to motivate the experiment and to deduce nonlocality, but my emphasis was on ''formal use made of'' it. No formula involves anything relativistic - only the talk around it does.

Precisely what "formal use" were you expecting? The motivation behind Bell's theorem is that relativity implies that the order of spacelike separated events is reference-frame dependent, so that the simplistic idea of causality we're used to breaks down if faster-than-light causal influences are allowed. Where the speed of light ends up in practice in Bell experiments is that it determines how stringent the timing of selection of detector settings and recording of outcomes has to be in order that it's the "no FTL causal influences" assumption that is being tested instead of something else. Bell doesn't spell all of this out because he assumes you, the reader, understand relativity and should find all of this obvious.

From my point of view, nothing more mysterious happens in Bell's experiment [...]

What you personally find intuitive or mysterious is subjective and not the issue here. You've made more specific claims that are unjustified, e.g., that Bell's theorem relies on a "particle" assumption and that it doesn't apply to classical electromagnetism.

[...] since exactly the same algebra is used [...]

So? Bell's theorem is not an exercise in pure mathematics. It is meant to say something about physics and, as such, the context in which Bell's algebra is applied matters.

 

[stevendaryl]

Why isn't there a fourth part $\gamma$, relevant to both Alice's and Bob's choice of detector setting? You make the assumption that this part is empty, but I cannot see a good reason for it. Thus $F_A$ and $F_B$ also depend on $\gamma$, and Bell's argument breaks down.

The point is that Alice's and Bob's choice of detector settings can be made at a spacelike separation. There is no reason to assume that their choices have anything in common. For example, let's suppose that each of them is carrying a little chunk of uranium, and they base their decision on which setting to choose on the number of decays (indicated by Geiger counter clicks) in a certain time interval. Then aren't those two choices completely independent? (At least, according to mainstream QM)

 

[N88]

The point is that Alice's and Bob's choice of detector settings can be made at a spacelike separation. There is no reason to assume that their choices have anything in common. For example, let's suppose that each of them is carrying a little chunk of uranium, and they base their decision on which setting to choose on the number of decays (indicated by Geiger counter clicks) in a certain time interval. Then aren't those two choices completely independent? (At least, according to mainstream QM)

IMO those two choices (say a and b) are completely independent. But, wondering if this next is an acceptable statement, I would like to add: The detectors are not independent.

The spacelike-separated detectors are correlated by a simple function of the two independent choices, the scalar product of a and b. So with C = a.b: C = +1 = parallel, C = 0 = orthogonal, C = -1 = anti-parallel, with physically-meaningful intermediate values. So IMO independent inputs do not deliver independent detectors when it comes to correlation. This seems to me to be a basis for understanding the correlation in "An abstract long-distance correlation experiment" before any experiment has been done.

 

[strangerep]

Modern relativity is the claim that Nature is ruled by Lorentz invariant laws, nothing else.

But I disagree with the "nothing else" part.

I'd have said that (special) relativity encompasses at least the Poincare group, restricted by some other empirically-motivated principles such as the (apparent) non-existence of tachyons, and +ve energy. I.e., causality is certainly part of modern relativity (else "modern relativity" would make false predictions).

 

[dextercioby]

Yes, modern relativity in the absence of gravity. It's one of the puzzles of physics how the Wigner theory (which starts with Poincaré invariance) leads to the theoretical possibility of tachyons, thus to a breach of the theory it started from.

 

[stevendaryl]

IMO those two choices (say a and b) are completely independent. But, wondering if this next is an acceptable statement, I would like to add: The detectors are not independent.

The spacelike-separated detectors are correlated by a simple function of the two independent choices, the scalar product of a and b. So with C = a.b: C = +1 = parallel, C = 0 = orthogonal, C = -1 = anti-parallel, with physically-meaningful intermediate values. So IMO independent inputs do not deliver independent detectors when it comes to correlation. This seems to me to be a basis for understanding the correlation in "An abstract long-distance correlation experiment" before any experiment has been done.

I'm not sure that I understand your point, but I'm not saying that there is no correlation between the detectors. What I'm trying to do is to factor the influences on the outcomes of the detectors into things that are shared between the two detectors, and the things that are not shared. So if the outcome at Alice's detector is determined (and that seems to be what A. Neumaier is saying--that if you could take into account the entire rest of the environment, that the outcomes become deterministic, in the same way that a coin flip is deterministic, if you only knew enough about breezes and the distribution of mass, etc.) and there are no FTL influences, then it would seem that Alice's result must be a deterministic function of what Alice and Bob share, plus what influences Alice alone. In my earlier post, \lambda represents everything that Alice and Bob had in common, and \alpha represents what was unique to Alice (e.g., her lump of uranium, if that's what she's using to pick her detector setting), and \beta represents what's unique to Bob (his lump of uranium). So if \alpha and \beta are independent, then that would seem to me to mean that Alice's result depends only on \alpha and \lambda, while Bob's result depends only on \beta and \lambda. It doesn't actually matter whether \lambda itself is local, or nonlocal. It only matters that Alice's result cannot depend on \beta and Bob's result cannot depend on \alpha.

 

[ddd123]

Instead of uranium they could be using light from distant galaxies from two antipodal regions of the sky. It's not far-fetched if you do the experiment on the ISS. Honestly, I think it drives superdeterminism to utter absurdity.

 

[zonde]

That (unlike $\lambda$, if I interpret its definition correctly) it is not determined by local information.

It does not matter how $\gamma$ is determined as long as it is independent from Alice's and Bob's measurement settings. Then it is shared information just like $\lambda$ .

It is a nontrivial restriction to assume that every piece of information must have originated at a single local point. It is this restriction that excludes nonlocal correlations.

There is no such assumption. There can be information that appears at two different locations independently. We even have a name for such information. It's called "coincidence".

 

[A. Neumaier]

What I'm trying to do is to factor the influences on the outcomes of the detectors into things that are shared between the two detectors, and the things that are not shared.

The possibility of such a separation assumes that the ''things'' are located at points. But it is precisely this idealization that plays havoc, already in classical relativity. In quantum field theory, it also causes initially problems (infinite interactions), which are then removed by renormalization. But renormalization turns point particles into point-like particles, which are (in principle infinitely) extended.

 

[A. Neumaier]

(special) relativity encompasses at least the Poincare group

Yes, special relativity encompasses the Poincare group, but relativity as such doesn't, as field theory in curved spaces (where the translation group is explictly broken but the local Lorentz groups are still intact) shows.

In quantum field theory we have to add unitaity and local commutativity, which automatically excludes tachyons. Classically, tachyons are not forbidden, and indeed Crenkov radiation is a tachyonic classical feature though it happens inside matter and not in vacuum.

 

[A. Neumaier]

There can be information that appears at two different locations independently.

But in extended objects the independentce is questionablle. The environment consists not of independent point objects.

 

[stevendaryl]

The possibility of such a separation assumes that the ''things'' are located at points. But it is precisely this idealization that plays havoc, already in classical relativity. In quantum field theory, it also causes initially problems 9infinite interactions), which are then removed by renrmalization. But renormalization turns point particles into point-like particles, which are (in principle infinitely) extended.

Well, let's look at the specific example I gave, namely the decay of uranium atoms. Alice has a blob of uranium. Bob has a blob of uranium. Yes, this situation has a description in terms of quantum field theory, with lepton and baryon fields. But are you saying that, because there is a description in terms of fields, the Geiger counter clicks at Alice and Bob are correlated in a nonlocal way?

 

[stevendaryl]

Well, let's look at the specific example I gave, namely the decay of uranium atoms. Alice has a blob of uranium. Bob has a blob of uranium. Yes, this situation has a description in terms of quantum field theory, with lepton and baryon fields. But are you saying that, because there is a description in terms of fields, the Geiger counter clicks at Alice and Bob are correlated in a nonlocal way?

Anyway, if it is true that the approximation that Alice's choice of detector settings and Bob's choice are not really independent, that seems like the superdeterminism loophole of Bell's theorem.