Commonsense local realism refutes Bell's theorem

[Mikko]

1) How many distinct lambdas are there in Bell's integral? (call that number N)

The integral is over all values of λ, as many as specified in the hidden variable model. Typically it is ℵ₁ but could be more or less.

2) Now if each particle pair measured has a distinct lambda, do you still think Bell's derivation follows?

Yes. If the probability of each value of λ is infinitesimal then each measurement has a distinct λ.

[gill1109]

1) How many distinct lambdas are there in Bell's integral? (call that number N)

The integral is over all values of λ, as many as specified in the hidden variable model. Typically it is ℵ₁ but could be more or less.

2) Now if each particle pair measured has a distinct lambda, do you still think Bell's derivation follows?

Yes. If the probability of each value of λ is infinitesimal then each measurement has a distinct λ.

Aleph? Which aleph? Typically lambda will be supposed to take values in some nice subset of a Euclidean space. For oractical purposes however, one could always take a very fine discretization and have a very large but finite sample space.

On the other hand: take a sample of size 10 000 from the standard normal distribution. Take another sample. All of the 2 x 10 000 values will certainly be different. Yet the averages of both sets of 10 000 will be very close to one another.

[Mikko]

Why should anyone try to disprove the QRC or find a 4xN table with CHSH correlation > 0.5? Can QM be used to make such a table?

No.

The Table would contain observable outcomes of a simulated experiment. So why cannot QM be used to populate such a table with observables, even where those QM calculations would be permitted to use formulae which relied on action at a distance?

Because such table cannot exist.

So why is it OK for QM to fail to make such a table of observables, but not OK for a hidden variable model to fail to make such a table?

From meanings of "local" and "realistic" it follows that a local realistic hidden variable model that mimics quantum mechanics can make such table. The wave function of Quantum Mechanics is not local, so there is no basis for a similar inference about QM.

[Mikko]

Aleph? Which aleph?

Aleph-1, ℵ₁, the cardinality of real numbers.

[minkwe]

1) How many distinct lambdas are there in Bell's integral? (call that number N)

The integral is over all values of λ, as many as specified in the hidden variable model. Typically it is ℵ₁ but could be more or less.

2) Now if each particle pair measured has a distinct lambda, do you still think Bell's derivation follows?

Yes. If the probability of each value of λ is infinitesimal then each measurement has a distinct λ.

Mikko, if each measurement has a distinct lambda Bell's inequalities relevant for any experiment can not be derived. This is clearly shown in Watson's paper and the thread I suggested above. Bell both assumes that the probabilities are infinitesimal and that the same lamdas are used in the P(a,b) measurement as in the P(a,c) measurement. Review his algebra leading up to eq 15.

The suggestion by some that it doesn't matter so long as the average value of lambda is the same, is just silly and not worthy of a response.

[minkwe]

From meanings of "local" and "realistic" it follows that a local realistic hidden variable model that mimics quantum mechanics can make such table.

Wrong. You are here again making the erroneous assumption that local-realism = 4×N. No EPRB experiment can ever make such a table, local or non-local. In other words, any such table if it exists can never represent outcomes of an EPRB experiment irrespective of the type of model producing the results ( local or non-local). The impossibility of producing the table is not a privilege of QM. It is a characteristic of the EPRB experiment itself. This was the point in Ben's question which you completely missed.

If each lambda is unique such a table can not exist, whether the model is local or non-local.

[gill1109]

1) How many distinct lambdas are there in Bell's integral? (call that number N)

The integral is over all values of λ, as many as specified in the hidden variable model. Typically it is ℵ₁ but could be more or less.

2) Now if each particle pair measured has a distinct lambda, do you still think Bell's derivation follows?

Yes. If the probability of each value of λ is infinitesimal then each measurement has a distinct λ.

Mikko, if each measurement has a distinct lambda Bell's inequalities relevant for any experiment can not be derived. This is clearly shown in Watson's paper and the thread I suggested above. Bell both assumes that the probabilities are infinitesimal and that the same lamdas are used in the P(a,b) measurement as in the P(a,c) measurement. Review his algebra leading up to eq 15.

The suggestion by some that it doesn't matter so long as the average value of lambda is the same, is just silly and not worthy of a response.

So Michel would you agree with Bell's derivation, if the hidden variable were discrete and we replaced integral .... rho(lamda) d lambda by sum ... p(lambda) ?

[gill1109]

The suggestion by some that it doesn't matter so long as the average value of lambda is the same, is just silly and not worthy of a response.

Nobody made such a suggestion.

[Ben6993]

Mikko ... thanks. That was helpful.

Why should any method of calculation be able to be used to construct an impossible table? Fred says the task is rigged (I think that his cynicism must be from being exposed to the problem longer than me)! QM says it is impossible. Michel's logic showed me that it is impossible in a 4xN table.

I am in danger of being seen as a heretic by the following, but I am sure someone will put me on the right path ...
my view is that the electron is only in spacetime at the two interactions (creation of the pair and measurement). I agree with QM that the particle is not in spacetime during the time of flight. That is why the QM calculations need extra dimensions. Ditto for a local hidden variable model, for which geometric algebra needs at least four spatial dimensions. QM and CA both need to do their calculations out of spacetime. Neither can make a 4xN table of outcomes with correl > 0.5.

I see two hidden variables (spin and angle) as being caused by physical properties of the electron. And those variables continue in the full dimensions of the particles. If an electron has a particular structure at creation giving it a particular angle, it will have that structure at later measurement, and would give the same angle inputted to the measurement. However the angle is not maintained through spacetime during the time of flight, but maintained by the many dimensions inhabited by the particle. That may not meet a definition of 'local'? That may not meet a definition of realism?

I see the hidden variables as 'local' because they are local to the particle, and part of its structure. I see the hidden variables as realistic because my own model of an electron is a concrete bunch of preons existing in more dimensions than 4. They are real enough to me but do not fit neatly into spacetime or into laboratory space. Except at interactions.

In other words, why cannot the hidden variables hide with the particle, wherever the particle is.

[gill1109]

By the way, integrals can be completely avoided in deriving CHSH.

There are just 16 different possible values of the quadruple (A(a, lambda), A(a', lambda), B(b, lambda), B(b', lambda)). Without loss of generality we can replace "lambda" by this element of {-1, +1}^4. All integrals reduce to sums over the 16 elementary possibilities.

[gill1109]

In other words, why cannot the hidden variables hide with the particle, wherever the particle is.

Because of Bell? Because, if you were right, then there would exist functions A(a, lambda) and ... and hence ... and hence your model could not reproduce the singlet correlations.

But maybe you don't believe in the singlet correlations.

Or maybe it is OK by you that Alice's choice of a influences not only the lambda of her particle but also that of the other particke.

[minkwe]

Ben,
The reason QM does not produce a 4×N table has nothing to do with non locality. It is the same reason why a LHV model of the EPRB experiment does not produce one -- is, it is impossible to measure a particle more than once. In QM terms, the E(a,b) measurement does not commute with the E(a,c) measurement on the same set of particles.

As Adenier explained, there are two possible interpretations for the QM E(a,b), E(a,c).
a) strongly objective: outcomes from a single set of particles ( the same set of lambdas in each expectation value).
b) weakly objective: outcomes from two separate sets of particles (different sets of lambdas)

Bell's inequalities can only be derived for the strongly objective case since you need the same set of lambdas in both for the algebra to proceed. Yet for the strongly objective case the QM expectations do not commute /are incompatible, so you can not simply add outcomes from two separate individual measurements and expect to get the same result as if you had measured both together.

By trivially adding the QM expectations as Bell does, when demonstrating violation by QM, he is assuming that they commute, which is the weakly objective interpretation. So the schism is due to erroneously mixing inequalities derived under a strongly objective interpretation with expectation values from a weakly objective interpretation.

This is the same error von Neumann made in his no-go theorem, which Bell described as silly.

Some here even go as far as to claim that there is no difference between the strongly objective and the weakly objective views. An idea I already demolished using a simple coin toss example(viewtopic.php?f=6&t=44#p1439), since it implies that counterfactual probabilities are the same as actual ones.

Note: All EPRB experimental results are necessarily weakly objective. The weakly objective inequalities have different upper bounds which have never been violated.

What Watson shows is that Bell's inequalities can not be derived under the weakly objective interpretation, which we must use if the inequalities should be relevant for actual experimental results.

[Heinera]

Why should anyone try to disprove the QRC or find a 4xN table with CHSH correlation > 0.5? Can QM be used to make such a table? The Table would contain observable outcomes of a simulated experiment. So why cannot QM be used to populate such a table with observables, even where those QM calculations would be permitted to use formulae which relied on action at a distance? So why is it OK for QM to fail to make such a table of observables, but not OK for a hidden variable model to fail to make such a table?

It is not about what is OK or not, but what is possible or not.

For QM theory, it is impossible to construct such a table for reasons inherent in the mathematical QM theory itself, quite unrelated to any experiments. And this impossibility is also why QM theory can generate the strong correlations, since these two properties are mathematically mutually incompatible.

With a LHV model, we can construct such a table, and there is a simple procedure to do so. It doesn't help if someone comes along and say "you really shouldn't be doing that, because it's inconsistent with QM." It can be done.

Take Joy's exploding balls experiment: Cameras record the angular momentum of a ball fragment. This value is later combined with an imagined detector angle to compute "up" or "down". Who says that it is impossible to apply different angles to one and the same angular momentum value, and do the computations all over again? Do we expect someone to come along and scream "Hey! Because in QM, an observation collapses the state, you can only do that computation once!"?

[harry]

1) How many distinct lambdas are there in Bell's integral? (call that number N)

The integral is over all values of λ, as many as specified in the hidden variable model. Typically it is ℵ₁ but could be more or less.

2) Now if each particle pair measured has a distinct lambda, do you still think Bell's derivation follows?

Yes. If the probability of each value of λ is infinitesimal then each measurement has a distinct λ.

Mikko, if each measurement has a distinct lambda Bell's inequalities relevant for any experiment can not be derived. This is clearly shown in Watson's paper and the thread I suggested above. Bell both assumes that the probabilities are infinitesimal and that the same lamdas are used in the P(a,b) measurement as in the P(a,c) measurement. Review his algebra leading up to eq 15.

The suggestion by some that it doesn't matter so long as the average value of lambda is the same, is just silly and not worthy of a response.

Indeed, Bell allows for as many lambda's as you want. And although I doubt that Bell's derivation follows, so far I have not identified any error with certainty. I'm still scrutinizing criticisms.
But in contrast with Watson's comments*, Bell assumed in his derivation that the same lambda's reoccur randomly (or effectively the same ones) for infinitely long series of detections at whatever angle, so that the same lambda occurs in each single line in his derivation. For it's the result that matters, and those results are recurring as they yield reproducible experiments. And that has nothing to do with an average value of lambda.

* As a reminder, Watson has: "from Bell’s own λ-license: λi =/= λn+i "

[gill1109]

As Adenier explained, there are two possible interpretations for the QM E(a,b), E(a,c).
a) strongly objective: outcomes from a single set of particles ( the same set of lambdas in each expectation value).
b) weakly objective: outcomes from two separate sets of particles (different sets of lambdas)

There is one and only one interpretation of the QM E(a, b): it stands for the probability two measured spins are equal minus the probability they are unequal. According to QM it equals <Psi | sigma(a) otimes sigma(b) |Psi> where
|Psi> = (|z+> otimes |z-> - |z-> otimes |z+> )/sqrt 2

If a local hidden variables theory would underly these predictions of QM, then one can give alternative expressions for E(a, b).

Adenier was, ten years ago, in my opinion, badly confused (I heard his talk and discussed it with him). His work on "weak and strong objectivity" was never published in peer reviewed journals and it has not had much of a following. Right now Adenier is working in main-stream quantum information theory so it appears he has "gone over to the other side". But I'll ask him next week in Vaxjo, where he nowadays stands.

Actually Adenier was just a main-stream Copenhagen interpretation person. There is nothing "behind" QM and it is counter-productive to look for something behind it.

However in quantum cryptography, as Adenier now well understands, it is important to ask if what you observe could be simulated classically, because if that is the case, what you observe is not cryptographically secure. Nowadays the LHV simulators (who all of course have to exploit one or the other loophole) are usefully employed in the quantum cryptography business. If they can't simulate it, it's likely secure. If they can, it certainly isn't.

This is nowadays big business, especially now that it looks as though the loopholes are close to overcome. We are close to a successful experiment with delayed choice and event ready detectors.

[gill1109]

But in contrast with Watson's comments*, Bell assumed in his derivation that the same lambda's reoccur randomly (or effectively the same ones) for infinitely long series of detections at whatever angle, so that the same lambda occurs in each single line in his derivation. For it's the result that matters, and those results are recurring as they yield reproducible experiments. And that has nothing to do with an average value of lambda.

* As a reminder, Watson has: "from Bell’s own λ-license: λi =/= λn+i "

What counts is the average value of A(a, lambda)B(b, lambda). What counts is that the result (this average value) does not depend on what measurenents, if any, are actually performed on the two partickes!

Ie the average of A(a, lambda)B(b, lambda) over many, many particle pairs is the same when a, b are measured; and when a, b' are measured; and when a' and b are measured...

[gill1109]

But in contrast with Watson's comments*, Bell assumed in his derivation that the same lambda's reoccur randomly (or effectively the same ones) for infinitely long series of detections at whatever angle, so that the same lambda occurs in each single line in his derivation. For it's the result that matters, and those results are recurring as they yield reproducible experiments. And that has nothing to do with an average value of lambda.

* As a reminder, Watson has: "from Bell’s own λ-license: λi =/= λn+i "

What counts is the average value of A(a, lambda)B(b, lambda). What counts is that the result (this average value) does not depend on what measurenents, if any, are actually performed on the two partickes!

Ie the average of A(a, lambda)B(b, lambda) over many, many particle pairs is the same when a, b are measured; and when a, b' are measured; and when a' and b are measured...

[Heinera]

It is impossible to measure a particle more than once. In QM terms, the E(a,b) measurement does not commute with the E(a,c) measurement on the same set of particles.

Don't you find it extremely ironic that you use a QM postulate ("It is impossible to measure a particle more than once") to argue against QM formalism and in favor of LHV models?

Take Joy's exploding balls experiment: Cameras record the angular momentum of a ball fragment. This value is later combined with an imagined detector angle to compute "up" or "down". Who says that it is impossible to apply different detector angles to one and the same angular momentum value, and do the computations all over again? Where does the QM postulate enter into all of that?

[Xray]

But in contrast with Watson's comments*, Bell assumed in his derivation that the same lambda's reoccur randomly (or effectively the same ones) for infinitely long series of detections at whatever angle, so that the same lambda occurs in each single line in his derivation. For it's the result that matters, and those results are recurring as they yield reproducible experiments. And that has nothing to do with an average value of lambda.

* As a reminder, Watson has: "from Bell’s own λ-license: λi =/= λn+i "

What counts is the average value of A(a, lambda)B(b, lambda). What counts is that the result (this average value) does not depend on what measurenents, if any, are actually performed on the two partickes!

Ie the average of A(a, lambda)B(b, lambda) over many, many particle pairs is the same when a, b are measured; and when a, b' are measured; and when a' and b are measured...

Gill, please clarify what you mean here. In some presentations on this subject, the FOUR (4) different orientations are Alice (a and a'), Bob (b and b').

Also, below Bell's (1964), equation (13), what is the meaning of "except at a set of points λ of zero probability."

[gill1109]

But in contrast with Watson's comments*, Bell assumed in his derivation that the same lambda's reoccur randomly (or effectively the same ones) for infinitely long series of detections at whatever angle, so that the same lambda occurs in each single line in his derivation. For it's the result that matters, and those results are recurring as they yield reproducible experiments. And that has nothing to do with an average value of lambda.

* As a reminder, Watson has: "from Bell’s own λ-license: λi =/= λn+i "

What counts is the average value of A(a, lambda)B(b, lambda). What counts is that the result (this average value) does not depend on what measurements, if any, are actually performed on the two partickes!

Ie the average of A(a, lambda)B(b, lambda) over many, many particle pairs is the same when a, b are measured; and when a, b' are measured; and when a' and b are measured...

Gill, please clarify what you mean here. In some presentations on this subject, the FOUR (4) different orientations are Alice (a and a'), Bob (b and b').

Also, below Bell's (1964), equation (13), what is the meaning of "except at a set of points λ of zero probability."

I don't know why Bell wrote "except at points of zero probability". Why waste time with this old-fashioined and obscure proof? Study my recent paper with a more simple proof of a more strong result.

The CHSH inequality has Alice choosing between settings a and a', and Bob between b and b'. So there are four pairs of settings: (a, b), (a, b'), (a', b), (a', b'). It is stronger than the original Bell inequality which has Alice choosing between settings a and b, and Bob between a and c. So four pairs: (a, a), (b, a), (a, c), (b, c). So this is a rather special set of four pairs of settings. Bell also assumes perfect anti-correlation (so the a,a correlation is -1). So he is restricting himself in two respects to special situations.