Local Causality and Bell's Second Theorem

[atyy]

That's a big ask, considering the array of evidence available. But I guess it is possible, that certainly was Einstein's view.

Dirac's too (not explicitly stated, but he does imply he expects new theories beyond quantum mechanics): "And when this new development occurs, people will find it all rather futile to have had so much of a discussion on the role of observation in the theory, because they will have then a much better point of view from which to look at things." https://blogs.scientificamerican.com/guest-blog/the-evolution-of-the-physicists-picture-of-nature/

 

[Elias1960]

"Realistic" in this context has a specific meaning, more or less the same as an "element of reality" per EPR (1935). For entangled Alice and Bob, that means:

Realism: Alice's X observable and Bob's non-commuting Y observable - either of which can be separately predicted with certainty - are both simultaneously well-defined even if they cannot both be simultaneously predicted in advance.

No. Realism, as far as necessary, is much weaker. It is the EPR criterion of reality. You have to add Einstein causality to be able to apply it to the experiment in question. Only Einstein causality, in a quite strong version (stronger than signal causality), allows making the conclusion that all the observables in question, given that they can be predicted without disturbing the system, have all definite values.

Usually one thinks that realism is what gives you the formula
$$E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,b,\lambda)B(a,b,\lambda)\rho(a,b,\lambda)d\lambda.$$
Then the rejection of superdeterminism reduces this to
$$ E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,b,\lambda)B(a,b,\lambda)\rho(\lambda)d\lambda$$
and Einstein causality reduces this to
$$ E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,\lambda)B(b,\lambda)\rho(\lambda)d\lambda,$$
which is already all you need to prove Bell's inequality.

Unfortunately for those who like to reject realism to save Einstein causality, this space $\Lambda$ can be constructed explicitly for quantum theory, the construction has been given in the paper where Kochen and Specker have proven their theorem:

Kochen, S., Specker, E.P. (1967). The Problem of Hidden Variables in Quantum Mechanics, J. Math. Mech. 17(1), 59-87

While this somewhat trivial construction is rejected as not defining what people think is worth to be named "hidden variables", it does not change the fact that the formulas don't care about what people think about that space $\Lambda$. Once that construction exists, however ugly, and you have Einstein causality and no superdeterminism, you can prove Bell's theorem. With this construction, realism reduces to nothing.

 

[Demystifier]

Of course, that is nowhere near what anyone means by anti-realistic. All you need to do to understand the "non-realistic" side of the street is accept that the Heisenberg Uncertainty Relations for a quantum system represents the state correctly for non-commuting observables. For example: There are not simultaneous well defined values for non-commuting P and Q. (I am ignoring interpretations here, as of course the Bohmian view is that they are both well defined but unknowable.)

The problem for realists is that if interpretations are ignored, then it's not clear what the Heinsenberg uncertainty means. Of course, it's clear what it means in the instrumental sense (the probability of such-and-such outcome under such-and-such measurement procedure is given by such-and-such formula ...), but for realists that's not what physics is really about. For realists physics is about the nature itself, not about measurements. For realists the measurement is just a tool to achieve the goal of understanding nature, while for instrumentalists the measurement itself is the goal.

So the real source of disagreement is not the interpretation of quantum mechanics, but the definition of physics. Realists accuse instrumentalists that they are engineers, not physicists. Instrumentalists accuse realists that they are philosophers, not physicists. As long as the physicists cannot agree what physics "is", or more generally what science "is", there will be unresolvable disagreements between realists and instrumentalists.

My "Bohmian mechanics for instrumentalists" is an attempt to find a middle ground.

 

[DrChinese]

No. Realism, as far as necessary, is much weaker. It is the EPR criterion of reality. You have to add Einstein causality to be able to apply it to the experiment in question. Only Einstein causality, in a quite strong version (stronger than signal causality), allows making the conclusion that all the observables in question, given that they can be predicted without disturbing the system, have all definite values.

Usually one thinks that realism is what gives you the formula
$$E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,b,\lambda)B(a,b,\lambda)\rho(a,b,\lambda)d\lambda.$$
Then the rejection of superdeterminism reduces this to
$$ E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,b,\lambda)B(a,b,\lambda)\rho(\lambda)d\lambda$$
and Einstein causality reduces this to
$$ E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,\lambda)B(b,\lambda)\rho(\lambda)d\lambda,$$
which is already all you need to prove Bell's inequality.

$$ E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,\lambda)B(b,\lambda)\rho(\lambda)d\lambda,$$
also implies - if we want realism/non-contextuality/observer independence in our Bell proof:
$$ E(AC|a,c) = \int_{\lambda\in\Lambda} A(a,\lambda)C(c,\lambda)\rho(\lambda)d\lambda,$$
$$ E(BC|b,c) = \int_{\lambda\in\Lambda} B(b,\lambda)C(c,\lambda)\rho(\lambda)d\lambda,$$

Bell does not show this step explicitly in his paper, but of course it is used after (14) as mentioned previously. It would be impossible to get Bell's result otherwise.

There is no question that EPR assumed strict Einstein causality - for EPR that meaning there is no spooky action at a distance of any kind. They would have been loathe to think otherwise, writing with the man who discovered special relativity. My 3 formulas are what EPR would have asked for if they wanted mathematical expressions of the 2 assumptions that match those implied (strong respect for relativity, i.e. that no influence propagates faster than c) or explicit in the EPR paper (that the EPR result depends on a reasonable view of objective reality, as mentioned in the second to the last paragraph). Bell implemented that view of objective reality in a way that would have satisfied EPR: It allows their view of Observer Independent Reality to be presented without requiring that it be experimentally provable. That would have therefore been "reasonable" to them.

That's Bell, which is all about answering EPR (as is clear from the title). Kochen Specker is a somewhat different animal, and I in no way want to lessen it. That's a thread of its own.

 

[Elias1960]

Bell implemented that view of objective reality in a way that would have satisfied EPR: It allows their view of Observer Independent Reality to be presented without requiring that it be experimentally provable. That would have therefore been "reasonable" to them.

That's Bell, which is all about answering EPR (as is clear from the title). Kochen Specker is a somewhat different animal, and I in no way want to lessen it. That's a thread of its own.

I disagree. Bell simply applied the EPR criterion of reality:

Now we make the hypothesis [2], and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of $\sigma_2$ , by previously measuring the same component of $\sigma_1$ , it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.

This is not a case of implementing something new, a view of reality or so, it is applying a published criterion of reality to a particular experimental situation.

Then, this criterion requires something which is experimentally provable, namely that one can predict the result of the experiment with certainty. That this happens in a way without disturbing the other states is, of course, an additional theoretical assumption. But there are no experimental facts without any theoretical background.

Last but not least, mentioning Kochen and Specker was not about their theorem, even if what is important here is part of the same paper:

Kochen, S., Specker, E.P. (1967). The Problem of Hidden Variables in Quantum Mechanics, J. Math. Mech. 17(1), 59-87

Given that the construction is quite simple, it needs only p.63.

 

[DrChinese]

1. I disagree. Bell simply applied the EPR criterion of reality:

Now we make the hypothesis [2], and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of σ2σ2 , by previously measuring the same component of σ1σ1 , it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.

This is not a case of implementing something new, a view of reality or so, it is applying a published criterion of reality to a particular experimental situation.

2. Then, this criterion requires something which is experimentally provable, namely that one can predict the result of the experiment with certainty. That this happens in a way without disturbing the other states is, of course, an additional theoretical assumption. But there are no experimental facts without any theoretical background.

1. The reasoning of the Bell passage you quoted is certainly good. A reasonable person would logically consider: if a measurement outcome can be predicted in advance without otherwise disturbing the system, then the result would be predetermined.

But clearly Bell added the new idea that all possible outcomes must be predetermined. EPR didn't need that extension to make the argument that QM was incomplete. All they needed was to prove that if a P (or Q) could be predicted with certainty, and QM could not do that, then it must be missing something.2. Bell couldn't have satisfied an experimental proof that each "element of reality" be predicted in advance simultaneously, because that is not possible. So he makes the assumption that for any and all pairs of angle setting, the statistical results (for counterfactual pairs) would not be affected by the choice of measurement setting. I.e he goes along with the following thinking of EPR:

Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depends upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this".

Bell then extends this to the calculation of the expectation value for pairs of angle settings (pairs ab, ac or bc in Bell), that counterfactual results could have been obtained and would not have been any different, hypothesizing them to be predetermined.

After reading this again, I am not sure we really disagree on anything concrete at this point. To me, KS proves that a non-contextual theory is not viable, local or not.* The point being that there is no direct contradiction between the Bell and KS theorems. Clearly they touch on many of the same ideas.

(*As I understand Bohmian Mechanics, it is contextual even though also nonlocal.)

(Edited for mistake regarding contextuality of BM.)

 

[Elias1960]

But clearly Bell added the new idea that all possible outcomes must be predetermined. EPR didn't need that extension to make the argument that QM was incomplete. All they needed was to prove that if a P (or Q) could be predicted with certainty, and QM could not do that, then it must be missing something.

This is not a new idea, but a logical consequence of "predetermined". The EPR criterion also formulates the EPR criterion using the counterfactual "if one can", not restricted to "if one does". So the conclusion "then there exists" does not depend on the measurement made in fact.

After reading this again, I am not sure we really disagree on anything concrete at this point. To me, KS proves that a non-contextual theory is not viable, local or not.* The point being that there is no direct contradiction between the Bell and KS theorems. Clearly they touch on many of the same ideas.

Again, my point is that Bell could have used the construction of KS p. 63 to show that

$$E(AB|a,b) = \int_{\lambda\in\Lambda} A(a,b,\lambda)B(a,b,\lambda)\rho(a,b,\lambda)d\lambda.$$

holds even in QM. So that the theorem is not at all about realism or so. The rejection of superdeterminism and Einstein causality would be sufficient.

(*As I understand Bohmian Mechanics, it is non-contextual even though also nonlocal.)

It is contextual. The results of non-configuration measurements depend on the configuration of the measurement device too, thus, on the context.

 

[DrChinese]

It is contextual. The results of non-configuration measurements depend on the configuration of the measurement device too, thus, on the context.

Sorry, I knew that and mis-typed. Demystifier had explained that to me some time back.

I consider QM to be a contextual theory. Which to me is the same thing as saying that observables do not have well defined values until measured. Which is the same thing as saying QM is not realistic, because all possible observables do not have well-defined values until measured. Yet Bohmians see it as realistic, which seems strange to me. Oh well, a subject for another thread...

 

[Elias1960]

I consider QM to be a contextual theory. Which to me is the same thing as saying that observables do not have well defined values until measured. Which is the same thing as saying QM is not realistic, because all possible observables do not have well-defined values until measured. Yet Bohmians see it as realistic, which seems strange to me. Oh well, a subject for another thread...

Maybe, but, first, contextuality appears in everyday life too. And even small children know this. Will my mother spend me some icecream? Usually one cannot know, and not because one simply does not know what she thinks about it, but simply because she has no opinion about this. One has to ask her to find out. And it matters much how one is asking her. If one asks her in a polite or rude way can heavily influence the result. Thus, the result is a contextual one.

Bohmians and other realists simply distinguish the configuration variables q(t) (which have definite values and follow a continuity equation) and all other variables that depend on the context.

BTW, this context-dependence of the other variables exists in classical mechanics too. All you have to look at is what depends on the trajectory q(t) alone and what depends on the Lagrange/Hamilton function, which contains all the information about the context. The configuration variables $q(t), \dot{q}(t)$ are not contextual. But the momentum already depends on the context of, say, an external EM field, with $p_i = \frac{\partial L}{\partial \dot{q^i}}= m \dot{q^i} + eA_i$ for the Lagrangian $L=\frac{1}{2m}\dot{q}^2 + e \dot{q}^i A_i$ .

 

[morrobay]

@Elias1960 , So you are very talented physicist: May I ask you for an explanation of the Bell inequality violations . Can be all inclusive, the assumptions, the logic, locality and realism. Thank you.

 

[Elias1960]

May I ask you for an explanation of the Bell inequality violations . Can be all inclusive, the assumptions, the logic, locality and realism. Thank you.

I think the explanation is the straightforward one - one introduces a preferred frame (which, in our universe, plausibly agrees with the CMBR frame). With a preferred frame, all the realistic interpretations of QM work nicely, neither realism nor causality poses any problem. And there is no reason at all to question classical logic as well as the classical objective Bayesian interpretation of probability (which Jaynes has named "logic of plausible reasoning").

Locality is also not really a problem, all that has to be rejected is Einstein locality. While QM itself is nonlocal, this may be an approximation, given that we have not yet identified the maximal speed of causal influences. So, the situation may be similar to that of Newtonian gravity: It is a theory with an infinite speed of information transfer, but it is now known to be an approximation for $c\to \infty$ of a theory of gravity with c as the limiting speed. This may happen again, we simply have not yet found the empirical evidence for the maximal speed of causal influences of quantum effects.

The formula which describes the faster than light causal influence is, in fact, part of the Schroedinger equation. If one rewrites it as two equations for $\psi(q) = \sqrt{\rho(q)} \exp \frac{i}{\hbar}\phi(q))$, the resulting equation for the probability density $\rho(q)$ is a continuity equation, that means, it contains an average velocity: $$\partial_t \rho(q,t) +\partial_i (\rho(q) v^i(q)) = 0$$ This velocity is local in the configuration space, that means, only the actual configuration matters, but the configuration is that of the global universe. And once the configuration is more than a single point particle, it follows that the velocity of change of the whole configuration depends on the whole configuration.

 

[WWGD]

Its sometimes called Bell Locality - non technically it means you have intuitive separate particles, technically its the usual condition of probabilistic Independence ie A theory θ is factorisable, i.e. satisfies factorisabilty, iff Pθ(A, B|a, b, c, λ) = Pθ(A|a, c, λ)Pθ(B|b, c, λ).

Thanks
Bill

Isnt this equality saying that events A,B ( under same conditions) are independent?

 

[PeterDonis]

Thread closed for moderation.

 

[bhobba]

Thread closed for moderation.

My query has been answered so will remain shut.

Thanks
Bill