Copenhagen: Restriction on knowledge or restriction on ontology?

[Demystifier]

"beables" (in normal language "observables")

Beables are not observables. The concept of beables was introduced by John Bell precisely to emphasize that they are different from observables. See e.g. the paper in my signature for more details.

 

[DarMM]

Nature lacks counterfactuals because that is what it means to be counterfactual. It is an epistemological concept that can never exist in actually measured experimental data.

Well certainly it lacks counterfactuals in the sense that if I put jam on my toast, then I didn't put marmalade on my toast, i.e. the event that didn't occur simply didn't occur. Only Many-Worlds lacks this factual definiteness as it is called in Quantum Foundations, since there all events occur.

However in the sense I mean here the point is that the variables you don't measure lack an outcome. This is distinct from classical mechanics where if I measure momentum it is still the case that there was a position value or if I only measure $L_z$ it is still the case that there were values for $(L_x, L_y, L_z)$. This is the counterfactual indefiniteness referred to here.

 

[DarMM]

What if the experiment is enriched with an additional piece of equipment that randomly decides whether the measurement will be performed in $d1$ or $d2$ direction? In this case all 4 outcomes are possible results of the measurement, so all 4 belong to the same sample space.

Yes, but the point is that there isn't an outcome of the form $(S_x, S_y, S_z)$ that's what is meant when it is said there isn't a single sample space. For example in your case the outcomes are:
$$
(heads, S_x = \frac{1}{2})\\
(heads, S_x = -\frac{1}{2})\\
(tails, S_z = \frac{1}{2})\\
(tails, S_z = -\frac{1}{2})
$$
So there is a single sample space, but not a single sample space for which the spin values are random variables.

 

[DarMM]

Any Bell inequality test that aims to close communication loophole has to combine all four experiments into one single experiment. Experiments like that have been performed number of times. Here is one https://arxiv.org/abs/quant-ph/9810080

So you are not making much sense to me.

Perhaps the example in the post above helps. So you can think of such an experiment as having a single sample space, but the point is that unlike classical mechanics $(S_x, S_y, S_z)$ don't all occur on each line. For instance if Spin were classical (even if it was stochastic) then for @Demystifier 's experiment we'd have something like (ignoring $S_y$):
$$
(heads, S_x = \frac{1}{2}, S_z = \frac{1}{2})\\
(heads, S_x = \frac{1}{2}, S_z = -\frac{1}{2})\\
(heads, S_x = -\frac{1}{2},S_z = \frac{1}{2})\\
(heads, S_x = -\frac{1}{2},S_z = -\frac{1}{2})\\
(tails, S_x = \frac{1}{2}, S_x = \frac{1}{2})\\
(tails, S_x = \frac{1}{2}, S_x = -\frac{1}{2})\\
(tails, S_x = -\frac{1}{2}, S_x = \frac{1}{2})\\
(tails, S_x = -\frac{1}{2}, S_x = -\frac{1}{2})
$$
And in this case you can extract the $\{heads, tails\}$ value and basically have a sample space of "facts" about the spin.

However in the quantum case in #178 you can't do this. There isn't a single sample space for the quantum observables and your experimental setting is sort of intrinsically embedded in the outcomes preventing you from isolating "facts" about the microscopic system.

You do have a single sample space, but it's not one for the properties of the subatomic system alone of the form you are able to construct in classical mechanics. It's a macroscopic setting and subatomic variable product space. So contextuality is still manifest and in fact this is a consequence of contextuality.

Does that make more sense?

 

[stevendaryl]

What if the experiment is enriched with an additional piece of equipment that randomly decides whether the measurement will be performed in $d1$ or $d2$ direction? In this case all 4 outcomes are possible results of the measurement, so all 4 belong to the same sample space.

And by the way, in the consistent histories interpretation, which perhaps is the most ontological version of "Copenhagen" interpretation, those 4 outcomes correspond to 4 histories that belong to the same consistency class.

That's basically what I said. I just don't understand what is meant when people talk about things not being from the same sample space.

Okay, what I do understand is that the Born rule cannot be taken as a probability distribution on "beables" to use Bell's phrase. If an electron is prepared to be spin-up in the z-direction, then you can't say "There is a 50% chance that it IS spin-up in the x-direction, and 50% chance that it IS spin-up in the y-direction". You can only say: "If I measure the spin in the x-direction, there is a 50% chance of getting 'up', and if I measure the spin in the y-direction, there is a 50% chance of getting 'up'". So I get that spins in the x-direction and spins in the y-direction don't belong to the same "probability space".

But it seems to me that if we include the choice and the result, we can certainly put them all into the same probability space: There is a 25% chance that I will measure the spin in the x-direction and get spin-up. There is a 25% chance that I will measure the spin in the x-direction and get spin-down. There is a 25% chance that I will measure the spin in the y-direction and get spin-up. There is a 25% chance that I will measure the spin in the y-direction and get spin-down.

 

[stevendaryl]

Beables are not observables. The concept of beables was introduced by John Bell precisely to emphasize that they are different from observables. See e.g. the paper in my signature for more details.

Yes, for Bell, a "beable" is some property that has a value whether it is observed, or not. Some physicists deny that there is such a thing, that nothing has a value unless it's measured. But that seems to me either to give a special role to consciousness or to lead to infinite regress: The particle doesn't have a position until it is detected, by (say), making the Geiger counter click. The Geiger counter doesn't click unless there is a microphone to record it. The microphone doesn't record it unless there is someone to play back the recording and hear it...

 

[stevendaryl]

Yes, for Bell, a "beable" is some property that has a value whether it is observed, or not. Some physicists deny that there is such a thing, that nothing has a value unless it's measured. But that seems to me either to give a special role to consciousness or to lead to infinite regress: The particle doesn't have a position until it is detected, by (say), making the Geiger counter click. The Geiger counter doesn't click unless there is a microphone to record it. The microphone doesn't record it unless there is someone to play back the recording and hear it...

Copenhagen seems to put macroscopic properties in the role of "beables". They have definite values at all times.

 

[DarMM]

I just don't understand what is meant when people talk about things not being from the same sample space.

But it seems to me that if we include the choice and the result, we can certainly put them all into the same probability space

Basically what is meant is that the quantum observables alone can't be put in a single sample space together. To have a single sample space you have to also include the actual device settings. This is directly related to or simply another variant of contextuality/lack of counterfactual definiteness and is the difference between QM and a classical stochastic theory.

Or as you said it's:
"If I measure the spin in the x-direction, there is a 50% chance of getting 'up', and if I measure the spin in the y-direction, there is a 50% chance of getting 'up'"

i.e. "chance if I measure...that I get" rather than "chance that it is". That's the difference between QM and a classical stochastic theory and is a direct consequence of contextuality/lack of counterfactual definiteness/lack of a single sample space for the observables themselves alone.

 

[zonde]

However in the quantum case in #178 you can't do this. There isn't a single sample space for the quantum observables and your experimental setting is sort of intrinsically embedded in the outcomes preventing you from isolating "facts" about the microscopic system.

You do have a single sample space, but it's not one for the properties of the subatomic system alone of the form you are able to construct in classical mechanics. It's a macroscopic setting and subatomic variable product space. So contextuality is still manifest and in fact this is a consequence of contextuality.

Does that make more sense?

I think there is problem with your choice of terminology. If you replace "single sample space" with "single set of initial conditions" it makes more sense. But it works only up to the point because then it would become quite clear that you are talking about superdeterminism, retrocausality or acausal constrain (there is correlation between initial conditions and measurement settings).

 

[DarMM]

I think there is problem with your choice of terminology.

Perhaps, but I don't get to choose it. It's the standard terminology.

If you replace "single sample space" with "single set of initial conditions" it makes more sense

"Single sample space for the quantum observables alone" is perhaps cleanest.

Just to note if you read quantum probability and quantum foundations texts they'll just say "no single sample space", but they mean "no single sample space for the quantum observables alone", i.e. they don't mention the possibility of enlarging the space to include the device settings since it wouldn't negate the main point.

 

[stevendaryl]

Or as you said it's:
"If I measure the spin in the x-direction, there is a 50% chance of getting 'up', and if I measure the spin in the y-direction, there is a 50% chance of getting 'up'"

i.e. "chance if I measure...that I get" rather than "chance that it is". That's the difference between QM and a classical stochastic theory and is a direct consequence of contextuality/lack of counterfactual definiteness/lack of a single sample space for the observables themselves alone.

I still don't get it. You can certainly concoct a classical model with the same probabilities as, say, the EPR experiment. Bell's theorem just tells us that you can't do it using local interactions.

For example:

1. Alice and Bob each submit their detector settings, $\alpha$ and $\beta$, respectively.
2. With probabilities $\frac{1}{2} sin^2(\frac{\theta}{2}), \frac{1}{2} sin^2(\frac{\theta}{2}), \frac{1}{2} cos^2(\frac{\theta}{2}), \frac{1}{2} sin^2(\frac{\theta}{2})$, we select one of the pairs: $(up, up), (down, down), (down, up), (up, down)$ (where $\theta = \beta - \alpha$)
3. Then Alice's result is the first element of the pair, and Bob's result is the second element of the pair.

That is implementable using classical stochastic processes. There is nothing quantum about it. And it has the same statistics as the quantum EPR experiment for anti-correlated spin-1/2 particles.

 

[zonde]

Perhaps, but I don't get to choose it. It's the standard terminology.

As far as I can trust wikipedia it is not standard terminology.
Sample space: In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment.

 

[DarMM]

You can certainly concoct a classical model with the same probabilities as, say, the EPR experiment. Bell's theorem just tells us that you can't do it using local interactions.

Well yes of course a nonlocal classical theory can do it, but this is the whole point of this thread.

To replicate the CHSH statistics you either have a classical stochastic theory which is nonlocal or what QM is, a generalized probability theory where there isn't a single sample space/the only single sample space intrinsically includes measurement settings, i.e. Participatory Realism where details about the observer are embedded in the sample space.

Those are the two different approaches to Bell's theorem, the first is the Bohmian form, the latter is the Copenhagen form.

My point was to point out the difference between QM and Classical Stochastic theories. It is contextuality/counterfactual indefiniteness in opposition to your claim that Contextuality had no foundational content. My point was never that classical processes can never, even if modified to being nonlocal, replicate the CHSH statistics.

 

[DarMM]

As far as I can trust wikipedia it is not standard terminology.
Sample space: In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment.

I'm saying that when Quantum Probability and Quantum Foundations texts and papers say "there is no single sample space" their meaning is the one I'm giving and that that choice of phrase is not a personal invention of mine but comes from those communities.

 

[zonde]

Yes, for Bell, a "beable" is some property that has a value whether it is observed, or not. Some physicists deny that there is such a thing, that nothing has a value unless it's measured. But that seems to me either to give a special role to consciousness or to lead to infinite regress: The particle doesn't have a position until it is detected, by (say), making the Geiger counter click. The Geiger counter doesn't click unless there is a microphone to record it. The microphone doesn't record it unless there is someone to play back the recording and hear it...

I would like to take a bit further this argument.
Let's suppose that we all agree that macroscopic systems have "beables". Now if microscopic systems don't have them where macroscopic systems get them?

You have made one point several times:

I think that most physicists are fairly confident that measuring devices themselves are described by the same quantum mechanics as the systems being studied.

If it's true I would say it is false confidence. Either you have to believe that microscopic systems have "beables" or you have to be able to explicitly demonstrate how apparent macroscopic "beables" can emerge from microscopic "beable-lessness" (and you should do that without any use of "bebles" obviously).

 

[Demystifier]

Copenhagen seems to put macroscopic properties in the role of "beables". They have definite values at all times.

Yes. The problem, of course, is that there is no sharp border between macro and micro. Hence, if macroscopic properties are beables, then some microscopic properties must be beables too. Such a point of view naturally leads to Bohmian mechanics as explained in the paper in my signature.

 

[stevendaryl]

Well yes of course a nonlocal classical theory can do it, but this is the whole point of this thread.

Okay, but every time the business about "different sample spaces" was brought up, it was said as if the probabilities themselves were incompatible with a stochastic process.

I would not say that this thread is about locality. It's about the meaning of the Copenhagen interpretation.

My point was to point out the difference between QM and Classical Stochastic theories.

But it doesn't seem to do that, since a classical stochastic theory can be contextual in the same way (which was what my example of implementing the EPR probabilities classically shows).

Maybe you can say that QM is different in that it implements contextuality of the sort in EPR using local means, but I don't see how you can say that. QM only gives the probabilities, not how they are implemented. The nonlocal stochastic model that I described is consistent with QM's predictions.

 

[rubi]

The situation is really simple: There is no state space $\Lambda$ such that $A_\alpha: \Lambda\rightarrow\{-1,1\}$ and $B_\beta: \Lambda\rightarrow\{-1,1\}$ are functions on this state space for every $\alpha$ and $\beta$ such that the correlations $\left<A_\alpha B_\beta\right>$ match the predictions of quantum mechanics. That's just a mathematical fact and there is just no way to circumvent it. It also has nothing to do with macroscopic or microscopic observables.

The only way to model the system on a single state space is to give up the idea that all $A_\alpha$ and $B_\beta$ are functions on the state space for all $\alpha$ and $\beta$. One has to admit, that this is a truly novel situation, which never occurs in classical theories.

 

[stevendaryl]

The situation is really simple: There is no state space $\Lambda$ such that $A_\alpha: \Lambda\rightarrow\{-1,1\}$ and $B_\beta: \Lambda\rightarrow\{-1,1\}$ are functions on this state space for every $\alpha$ and $\beta$ such that the correlations $\left<A_\alpha B_\beta\right>$ match the predictions of quantum mechanics. That's just a mathematical fact and there is just no way to circumvent it. It also has nothing to do with macroscopic or microscopic observables.

The only way to model the system on a single state space is to give up the idea that all $A_\alpha$ and $B_\beta$ are functions on the state space for all $\alpha$ and $\beta$. One has to admit, that this is a truly novel situation, which never occurs in classical theories.

But as I have said a number of times, there is no problem getting the probabilities to work out the same as EPR using classical probabilities if we allow nonlocal interactions. So the correct conclusion, as Bell said, is that it's a situation which never occurs in LOCAL classical theories.

 

[Demystifier]

But as I have said a number of times, there is no problem getting the probabilities to work out the same as EPR using classical probabilities if we allow nonlocal interactions. So the correct conclusion, as Bell said, is that it's a situation which never occurs in LOCAL classical theories.

I think you and @rubi are using the word "classical" with different meanings.

 

[DarMM]

Okay, but every time the business about "different sample spaces" was brought up, it was said as if the probabilities themselves were incompatible with a stochastic process.

No, I was saying that the differences between QM and classical stochastic processes was in the contextuality/counterfactual indefiniteness.

This was mainly in response to you claiming, counter to the opinion of most experts, that contextuality had no foundational import. I certainly never claimed (or intended to claim, perhaps in the ream of posts I got muddled somewhere) that classical stochastic processes cannot replicate CHSH probabilities as from Bohmian Mechanics they clearly can.

I would not say that this thread is about locality. It's about the meaning of the Copenhagen interpretation.

@Demystifier was basically asking about two things. Exactly what Copenhagen and similar views mean by "know" and exactly how they get around nonlocality or is their method for getting around nonlocality consistent. I answered the first part in my initial post and the rest has been about the second point.

 

[rubi]

But that's not true. As I have said a number of times, there is no problem getting the probabilities to work out the same as EPR using classical probabilities if we allow nonlocal interactions.

Well, then please provide a counterexample. It should look something like this:
$\Lambda = \{x,y,z\}$
$A_0(x) = 1, A_0(y) = -1, A_0(z) = -1$
$A_1(x) = -1, A_1(y) = 1, A_1(z) = -1$
(and so on for $B_\beta$).
Then provide a probability distribution on $\Lambda$, e.g.
$P(x) = 0.5, P(y) = 0.5, P(z) = 0$
And then calculate the correlations $\left<A_\alpha B_\beta\right>$ and compare them to the QM result. You won't be able to do this (if you want them to match).

 

[DarMM]

But it doesn't seem to do that, since a classical stochastic theory can be contextual in the same way (which was what my example of implementing the EPR probabilities classically shows).

A classical stochastic theory can replicate the statistics, but it doesn't need to either:

1. Give up a common sample space
2. Obtain a common sample space at the cost of embedding the observer in some form

It avoids both with a nonlocal $\lambda$, so it is not contextual in the same way.

 

[stevendaryl]

Well, then please provide a counterexample. It should look something like this:
$\Lambda = \{x,y,z\}$
$A_0(x) = 1, A_0(y) = -1, A_0(z) = -1$
$A_1(x) = -1, A_1(y) = 1, A_1(z) = -1$
(and so on for $B_\beta$).
Then provide a probability distribution on $\Lambda$, e.g.
$P(x) = 0.5, P(y) = 0.5, P(z) = 0$
And then calculate the correlations $\left<A_\alpha B_\beta\right>$ and compare them to the QM result. You won't be able to do this (if you want them to match).

I'm not disputing Bell's theorem. But his conclusion was not that there was no classical probabilistic model that could reproduce the predictions of EPR. His conclusion was that there was no LOCAL model using classical probabilities that could reproduce the predictions of EPR. The probabilities in the EPR model can be reproduced by a classical stochastic model, if you allow nonlocal interactions. So the EPR statistics don't say anything about classical versus nonclassical unless you impose locality.

 

[rubi]

I'm not disputing Bell's theorem. I'm saying that the probabilities in the EPR model can be reproduced by a classical stochastic model, if you allow nonlocal interactions. So the EPR statistics don't say anything about classical versus nonclassical unless you impose locality.

But your stochastic model will still not have the functions $A_\alpha$ and $B_\beta$. One has to admit that this is a true novelty compared to classical mechanics.

 

[stevendaryl]

But your stochastic model will still not have the functions $A_\alpha$ and $B_\beta$. One has to admit that this is a true novelty compared to classical mechanics.

My model is implemented using classical mechanics. The difference with quantum mechanics is the timing: Bob's result has to be at a timelike separation from Alice's choice of measurements, while quantum mechanically, it can be at a spacelike separation.

The functions $A_\alpha$ and $B_\beta$ are only relevant if you are trying to give a local hidden-variables model that reproduces the EPR predictions.

 

[rubi]

My model is implemented using classical mechanics. The difference with quantum mechanics is the timing: Bob's result has to be at a timelike separation from Alice's choice of measurements, while quantum mechanically, it can be at a spacelike separation.

It is certainly possible to arrange for an experiment such that given a time slicing of spacetime, Alice and Bob perform their measurements simultaneously with respect to this time slicing. The theorem I quoted holds in this situation. Are you saying that your model is unable to reproduce the correlations in such a situation?

The functions $A_\alpha$ and $B_\beta$ are only relevant if you are trying to give a local hidden-variables model that reproduces the EPR predictions.

No, it has nothing to do with hidden variables. Every Kolmogorovian probability theory needs a probability space $\Lambda$ and observables modeled by functions on this space. If you relax this requirement, you're not dealing with classical probability theory or classical stochastic processes anymore. Whether this space represents a space of physical hidden variables or not is irrelevant.

 

[A. Neumaier]

It is certainly possible to arrange for an experiment such that given a time slicing of spacetime, Alice and Bob perform their measurements simultaneously with respect to this time slicing.

Even stronger, if they perform their measurement at spacelike separation, there is always a frame in which they perform it simultaneously.

 

[zonde]

Even stronger, if they perform their measurement at spacelike separation, there is always a frame in which they perform it simultaneously.

Simultaneity in special relativity is only a convention and it does produce any physical consequences.

 

[A. Neumaier]

Simultaneity in special relativity is only a convention and it doesn't produce any physical consequences.

But the wave function and its probability interpretation depends on the frame. Thus anything argued with in these terms depends on it, too.

Note that we are not arguing about physical consequneces but about how one talks about these.

 

[DarMM]

No, it has nothing to do with hidden variables. Every Kolmogorovian probability theory needs a probability space $\Lambda$ and observables modeled by functions on this space. If you relax this requirement, you're not dealing with classical probability theory or classical stochastic processes anymore. Whether this space represents a space of physical hidden variables or not is irrelevant.

I made this point in post #146 where @stevendaryl replied with a card analogy. Maybe take a look at that since you seem to be explaining things in a much cleaner fashion than I was.

 

[zonde]

No, it has nothing to do with hidden variables. Every Kolmogorovian probability theory needs a probability space $\Lambda$ and observables modeled by functions on this space. If you relax this requirement, you're not dealing with classical probability theory or classical stochastic processes anymore. Whether this space represents a space of physical hidden variables or not is irrelevant.

Probability space does not contain state space. And it does not contain functions from state space to outcomes. It only contains functions from outcomes to probabilities. That's the very idea of probabilities that outcomes are treated as uncertain.

 

[DarMM]

Probability space does not contain state space. And it does not contain functions from state space to outcomes. It only contains functions from outcomes to probabilities. That's the very idea of probabilities that outcomes are treated as uncertain.

A Probability space $\Omega$ has a measure, which is a function from Events (subsets of $\Omega$) to the Reals (their probability):
$$\mu : E \rightarrow \mathbb{R}\\
E \subset \Omega$$
You can't always assign probabilities to outcomes although in the finite case you can. You then have random variables which are functions on $\Omega$:
$$X : \Omega \rightarrow \mathbb{R}$$
The latter is what @rubi is talking about. It's how observables are formulated.

 

[zonde]

But the wave function and its probability interpretation depends on the frame. Thus anything argued with in these terms depends on it, too.

Note that we are not arguing about physical consequneces but about how one talks about these.

Any non-local treatment requires you to pick preferred reference frame. It is not supposed to work in any reference frame the way it works in preferred reference frame.

 

[vanhees71]

It depends on what one means by "satisfactorily". I am quite satisfied with the idea proposed in my signature below.

Satisfactory means it reinterprets the Standard Model in a Bohmian way. I've not seen this achieved yet. Your paper is nice but there are many words and not a worked-out mathematical description!