Thermal interpretation and Bell's inequality

[DrChinese]

#10. The corollary being that, when we avoid this breach, we can derive valid inequalities: doubly valid because they agree with QM, and with locality.

That is not correct. There are 2 assumptions in Bell, locality/separability being one of them. Bell inequality violation indicating that at least one is incorrect. It does not say which one(s).

 

[N88]

I don't think you can say that because the existence of correlations that violate the Bell inequalities suggests that there are nonlocal "beables" that would violate your statement here. In other words, such correlations suggest that there has to be something connecting spacelike separated measurements, even if that something can't be used to send actual information faster than light (i.e., signal locality holds).
How can that be since it's directly observable? We have experimentally confirmed that you can get correlations that violate the Bell inequalities, and that's what "nonlocality" means.

But aren't the Bell inequalities based on naive-realism?

By which I mean views like this:

Realism is the opinion that objects have values for observable properties: and that these values exist prior to measurement and independent of the choice of measurement (ie, they are noncontextual).

I ask because [if true] such would appear (to me) to be so naive as to be dismissed out of hand? And to thus have no relevance to locality?

PS: What is your definition, please, of the realism that underpins Bell's analysis?

Thanks

 

[PeterDonis]

aren't the Bell inequalities based on naive-realism?

They're based on the assumptions Bell specifically gave in his paper. Those assumptions are stated in math, not ordinary language. What ordinary language people choose to describe those assumptions is a matter of words, not math or physics.

What is your definition, please, of the realism that underpins Bell's analysis?

I don't have one. I haven't used the word "realism", and I am not going to try to give a definition of a vague ordinary language term that has nothing to do with the actual math or physics.

 

[N88]

That is not correct. There are 2 assumptions in Bell, locality/separability being one of them. Bell inequality violation indicating that at least one is incorrect. It does not say which one(s).

Please see my last response, Post #37: Isn't a naivish realism also an assumption in Bell's work?

How do you define Bell's "realism" please?

 

[N88]

They're based on the assumptions Bell specifically gave in his paper. Those assumptions are stated in math, not ordinary language. What ordinary language people choose to describe those assumptions is a matter of words, not math or physics.
I don't have one. I haven't used the word "realism", and I am not going to try to give a definition of a vague ordinary language term that has nothing to do with the actual math or physics.

Thank you. I like this approach, and appreciate you taking such a clear position. Especially as it seems to me that realism [a term to me so confusedly used] "of some sort" gives rise to this question:

Could you explain please (in your terms) "the physics" (shall we call it) behind Bell's move from the first equation on p.198 [of his famous essay] to the second equation?

PS: I see that he uses his eqn (1), and thus (I presume) its conditions about "instances". But his "physics" must allow him to somehow merge the 2 instances in p.198's first eqn to deliver different instances in p.198's second eqn. Thanks.

 

[DarMM]

Could you explain please (in your terms) "the physics" (shall we call it) behind Bell's move from the first equation on p.198 [of his famous essay] to the second equation?

Do you mean the two different forms of the integral prior to the inequality?

It's just a simple reordering of the terms given they only have values $\pm 1$

 

[N88]

(X). Do you mean the two different forms of the integral prior to the inequality?

(Y). It's just a simple reordering of the terms given they only have values $\pm 1$

Yes to (X) and the difference between the first and second integral on p.198. But I think (Y) is invalid.

To be clearer: I am seeking to understand the "physics" that Bell has in mind when using his eqn (1) to move from the 2 instances in p.198's first eqn to the 2 instances in p.198's second eqn. It seems to me that Bell's key "physics" assumption -- and where he departs from the related physical-reality that he is studying; since his departure is violated by QM -- can only be introduced in this move: everything else is plain mathematics.

You are correct that these instances can only deliver $\pm 1$. But please note that his eqn (1) from p.196 is qualified by the sentence that introduces it. So the EPRB correlations -- that Bell is working with when he uses his eqn (13) in the derivation of his inequality -- would not hold if A and B are put together from different instances simply by a reordering.

 

[DarMM]

I don't get what you mean, he's just using the fact that they equal $\pm 1$ and a bit of simple algebra.

 

[N88]

I don't get what you mean, he's just using the fact that they equal $\pm 1$ and a bit of simple algebra.

OK, thanks, let me try again. Let's call the 2 top equations on p.198: (14a) and (14b).

So, to me, (14a) is true; it is simply a definition of Bell's terms.

But (14b) looks false to me; and this view is backed by the fact that it (by plain mathematics) leads to Bell's famous eqn (15): which IS false under QM.

So I'm trying to understand: what is Bell up to -- mentally, physically, mathematically, if it helps --- when he moves from (14a) to (14b)?

For to me, whatever he's done in that single step, it is not correct under QM or EPRB nor mathematically: AND I still cannot see any reasonable way to make it work (though several advisors here have surely tried).

 

[PeterDonis]

Could you explain please (in your terms) "the physics" (shall we call it) behind Bell's move from the first equation on p.198 [of his famous essay] to the second equation?

It's not physics, it's math. Bell does not give any specific physical interpretation of his math. He just does the math.

whatever he's done in that single step, it is not correct under QM or EPRB

Of course not; the whole point of the paper is to show that the mathematical model he is using cannot match the predictions of QM. As I've already pointed out multiple times. I cannot understand why you keep saying his model doesn't match QM as if it were a problem, when it's precisely the point of his paper.

nor mathematically

It's perfectly correct mathematically; I've already demonstrated that in post #14.

this view is backed by the fact that it (by plain mathematics) leads to Bell's famous eqn (15): which IS false under QM

This argument is wrong, and I don't understand why you would even make it. Once more: the fact that Bell's mathematical model makes predictions which do not match QM is the entire point of the paper. Bell is not trying to construct a mathematical model that will match the predictions of QM. He is trying to show that a certain class of mathematical models, the ones that satisfy his assumptions, cannot match the predictions of QM. That is the entire point of the paper.

Please read the above again and again until it sinks in. This whole thread is basically going in circles because you have not grasped this basic point.

 

[N88]

It's not physics, it's math. Bell does not give any specific physical interpretation of his math. He just does the math.
Of course not; the whole point of the paper is to show that the mathematical model he is using cannot match the predictions of QM. As I've already pointed out multiple times. I cannot understand why you keep saying his model doesn't match QM as if it were a problem, when it's precisely the point of his paper.
It's perfectly correct mathematically; I've already demonstrated that in post #14.
This argument is wrong, and I don't understand why you would even make it. Once more: the fact that Bell's mathematical model makes predictions which do not match QM is the entire point of the paper. Bell is not trying to construct a mathematical model that will match the predictions of QM. He is trying to show that a certain class of mathematical models, the ones that satisfy his assumptions, cannot match the predictions of QM. That is the entire point of the paper.

Please read the above again and again until it sinks in. This whole thread is basically going in circles because you have not grasped this basic point.

Sorry for my slowness; I share your frustration. Did you reply to post #15. That will help me to see how Bell's mathematical model departs from QM and nature. Thanks.

 

[PeterDonis]

Did you reply to post #15.

Yes, in post #21. To repeat my reply there: QM (and any interpretation of QM, including TI) has absolutely nothing to say about Bell's math--the whole derivation in the paper, not just the equations at the top of p. 198, the whole thing. Bell's entire derivation has nothing whatever to do with QM. He is just making some mathematical assumptions and deriving their consequences. That derivation has nothing to do with QM and QM says nothing about it. The only role QM plays in any of this is that its predictions are different from the ones Bell gets from the mathematical model he constructs. That's it.

 

[PeterDonis]

QM (and any interpretation of QM, including TI) has absolutely nothing to say about Bell's math

Perhaps it might help if I try to pose a different question, one that I think you should be asking instead of the ones you are asking:

Why do people think Bell's derivation is such a big deal?

I think the reason is that, before Bell wrote his paper, practically all physicists had implicitly assumed, without really thinking about it, that any physical theory would be capable of being written mathematically in a way that would satisfy Bell's assumptions. In particular, the two that have been pointed out in this thread: that probabilities of correlations would always factorize, as in Bell's equation (2) (note that the A probability in the integral there does not depend on the $\vec{b}$ setting, and vice versa), and that it would always be possible to introduce "counterfactual" items into the equations, such as the vector $\vec{c}$ that is introduced in Bell's equation at the top of p. 198 (I call this vector "counterfactual" because it is different from either of the measurement settings that are actually measured), and still derive valid predictions from them.

But Bell showed that any mathematical model that satisfies those assumptions cannot match the predictions of QM, and since the predictions of QM are correct for these experiments (as has now been verified many times), physicists now are forced to grapple with the fact that the implicit assumptions they had been making about how valid physical theories could be expressed mathematically are wrong. And physicists find it very difficult to imagine how they could be wrong--how Nature could work in such a way that a model that satisfies such seemingly simple and innocuous assumptions cannot match actual experimental results.

All this is true, but it does not mean that QM itself has anthing to say about Bell's mathematical model and its consequences, other than the obvious fact that QM's predictions are different. Bell's model is simply a different theory from QM (or rather a general class of theories all of which are different from QM), which we now know is falsified by Nature. That's all there is to it.

 

[DrChinese]

Please see my last response, Post #37: Isn't a naivish realism also an assumption in Bell's work?

How do you define Bell's "realism" please?

EPR defined "elements of reality". Then they asked whether those elements were simultaneously real if they could not be simultaneously measured. They felt it unreasonable to require them to simultaneously measured to be real. In other words, they assumed counterfactual definiteness. I would call that realism. I don't call this "naive" as I don't know what useful definition of realism exists which is "less naive".

 

[PeterDonis]

they assumed counterfactual definiteness

Which, just to relate this back to the math in Bell's paper, is the assumption that introducing the extra vector $\vec{c}$ at the top of p. 198 is valid, even though that vector does not describe either of the measurements that are actually made.

The other assumption, that the joint probability factorizes as in Bell's equation (2), is the one that is usually called "locality".

 

[DarMM]

OK, thanks, let me try again. Let's call the 2 top equations on p.198: (14a) and (14b).

So, to me, (14a) is true; it is simply a definition of Bell's terms.

But (14b) looks false to me; and this view is backed by the fact that it (by plain mathematics) leads to Bell's famous eqn (15): which IS false under QM.

So the first equation has:
$$-A(a, \lambda)A(b, \lambda) + A(a, \lambda)A(c, \lambda)$$
Extract $A(a, \lambda)$:
$$A(a, \lambda)\left[A(c, \lambda) - A(b, \lambda)\right]$$
Since $A(b,\lambda)^{2} = 1$ by the equation (1) in the paper:
$$A(a, \lambda)\left[A(b,\lambda)^{2}A(c, \lambda) - A(b, \lambda)\right]$$
Then just extract $A(b,\lambda)$:
$$A(a, \lambda)A(b,\lambda)\left[A(b,\lambda)A(c, \lambda) - 1\right]$$

So it's just basic algebra. No physical assumptions. The assumption QM breaks comes earlier in the paper. It's the assumption that $A, B, C$ are in fact random variables over some common space of polarization settings and variables $\lambda$.

 

[N88]

So the first equation has:
$$-A(a, \lambda)A(b, \lambda) + A(a, \lambda)A(c, \lambda)$$
Extract $A(a, \lambda)$:
$$A(a, \lambda)\left[A(c, \lambda) - A(b, \lambda)\right]$$
Since $A(b,\lambda)^{2} = 1$ by the equation (1) in the paper:
$$A(a, \lambda)\left[A(b,\lambda)^{2}A(c, \lambda) - A(b, \lambda)\right]$$
Then just extract $A(b,\lambda)$:
$$A(a, \lambda)A(b,\lambda)\left[A(b,\lambda)A(c, \lambda) - 1\right]$$

So it's just basic algebra. No physical assumptions. The assumption QM breaks comes earlier in the paper. It's the assumption that $A, B, C$ are in fact random variables over some common space of polarization settings and variables $\lambda$.

But can't we test your basic algebra by stopping at the second equation and completing the integral there?

Since the result is nothing like Bell's inequality, it looks like Bell's difficulties are already in the second equation?

 

[PeterDonis]

can't we test your basic algebra by stopping at the second equation and completing the integral there?

Since the result is nothing like Bell's inequality

Please show your work. It might not look like Bell's inequality to you, but if you want to claim it's mathematically inconsistent with Bell's inequality, in the face of two people now showing you the basic algebra that converts on integrand into the other, you are going to have to show us the math in detail. Either that or we can just close this thread since obviously you are simply refusing to accept what anyone tells you.

 

[N88]

Please show your work. It might not look like Bell's inequality to you, but if you want to claim it's mathematically inconsistent with Bell's inequality, in the face of two people now showing you the basic algebra that converts on integrand into the other, you are going to have to show us the math in detail. Either that or we can just close this thread since obviously you are simply refusing to accept what anyone tells you.

Please, I am not seeking to refuse anything. I am seeking to understand what I am being told. I understood you to say that Bell's inequality is based on mathematics unrelated to QM.

In asking me to show my work in detail: by observation it appears that completion of the integral over DarMM's second equation equals zero? Reason: The first term takes the values ±1 randomly, so (under integration over λ) its average value will be zero.

That was all.

 

[PeterDonis]

I understood you to say that Bell's inequality is based on mathematics unrelated to QM.

That's correct.

by observation it appears that completion of the integral over DarMM's second equation equals zero? Reason: The first term takes the values ±1 randomly, so (under integration over λ) its average value will be zero.

If this is true of any version of the integral, it's true of all of them; all of the integrands are simple algebraic transformations of each other, so they must all result in the same value on integration. So your reasoning here must be wrong.

 

[N88]

N88 said: I understood you to say that Bell's inequality is based on mathematics unrelated to QM.

That's correct.

N88 said: by observation it appears that completion of the integral over DarMM's second equation equals zero? Reason: The first term takes the values ±1 randomly, so (under integration over λ) its average value will be zero.

If this is true of any version of the integral, it's true of all of them; all of the integrands are simple algebraic transformations of each other, so they must all result in the same value on integration. So your reasoning here must be wrong.

Are you in a position to help me find my error?

It looks like a straight-forward integral whose first term (a function of λ) takes the values ±1 randomly?

 

[PeterDonis]

Are you in a position to help me find my error?

No, since you're unwilling to do even the simple work of writing down the integral in question and explaining, term by term, what you think it should give and why. Either show your work or this thread will be closed.

 

[N88]

No, since you're unwilling to do even the simple work of writing down the integral in question and explaining, term by term, what you think it should give and why. Either show your work or this thread will be closed.

From the second eqn in Post #51 above, completing the integral in line with Bell (1964):​

$$\smallint d\lambda \rho(\lambda)A(a, \lambda)\left[A(c, \lambda) - A(b, \lambda)\right]=0$$

since $A(a, \lambda)$ is a function of $\lambda$ and takes the values ±1 randomly.​

 

[PeterDonis]

since $A(a, \lambda)$ is a function of $\lambda$ and takes the values ±1 randomly

But the integrand is not just $A(a, \lambda)$. It contains two other factors, both of which also vary with $\lambda$. You have to look at how the entire integrand varies with $\lambda$, not just one factor.

 

[Elias1960]

Bell assumes no retrocausality, no nonlocal effects, all variables come from a common sample space and that there is a single world.

In his theorem.

Beyond the theorem, he was (at that time essentially the only one) supporter of Bohmian mechanics, which is nonlocal, and the aim of the theorem was to prove that the nonlocality of BM is not a valid argument against it, because it is unavoidable for any realistic interpretation of quantum theory.

 

[N88]

For PeterDonis: Of course! Many thanks and my apologies! The problem occurs lower down! Let me redefine it using {.} to keep related terms together:

From the first eqn in Post #51 above, including the integrals in line with Bell (1964):​

$$E(a,b)-E(a,c)=-\smallint d\lambda \rho(\lambda)\left[\{A(a, \lambda)A(b, \lambda)\}-\{A(a, \lambda)A(c, \lambda)\}\right]\;\;(1)$$
$$=-\smallint d\lambda \rho(\lambda)\{A(a, \lambda)A(b, \lambda)\}\left[1-\{A(a, \lambda)A(b, \lambda)\}\{A(a, \lambda)A(c, \lambda)\}\right].\;\;(2)$$
$$So, since \;A(a, \lambda)A(b, \lambda)\leq1: \;\;(3)$$
$$ |E(a,b)-E(a,c)|\leq\smallint d\lambda \rho(\lambda)\left[1-\{A(a, \lambda)A(b, \lambda)\}\{A(a, \lambda)A(c, \lambda)\}\right]\;\;(4)$$
$$\leq1-E(a,b)E(a,c).\;\;(5)$$

Now, in eqn (4), if I used $A(a, \lambda)A(a, \lambda)=1$, I would get Bell's inequality; just like you and DarMM and many others do using $A(b, \lambda)A(b, \lambda)=1$ in the alternative formulation in Post #51.

But then I would have converted (5) -- by an improper separation of terms* -- from an inequality that is never false into Bell's inequality which is frequently false.

* By an improper separation I mean this: I would have converted $E(a,b)E(a,c)$ into $E(b,c)$; which is unlikely to be true. That a function $F(b,c)$ should equal a function $F(a,b)F(a,c)$ over all $a,b,c$.

The above should show why I am still seeking to understand the mathematics behind Bell's theorem.

My question: Given the above, does Bell's inequality $|E(a,b)-E(a,c)|\leq1+E(b,c)$ arise from an improper mathematical separation of terms?

​

 

[DarMM]

From the first eqn in Post #51 above, including the integrals in line with Bell (1964):
$$E(a,b)-E(a,c)=-\smallint d\lambda \rho(\lambda)\left[\{A(a, \lambda)A(b, \lambda)\}-\{A(a, \lambda)A(c, \lambda)\}\right]\;\;(1)$$
$$=-\smallint d\lambda \rho(\lambda)\{A(a, \lambda)A(b, \lambda)\}\left[1-\{A(a, \lambda)A(b, \lambda)\}\{A(a, \lambda)A(c, \lambda)\}\right].\;\;(2)$$
$$So, since \;A(a, \lambda)A(b, \lambda)\leq1: \;\;(3)$$
$$ |E(a,b)-E(a,c)|\leq\smallint d\lambda \rho(\lambda)\left[1-\{A(a, \lambda)A(b, \lambda)\}\{A(a, \lambda)A(c, \lambda)\}\right]\;\;(4)$$
$$\leq1-E(a,b)E(a,c).\;\;(5)$$

Now, in eqn (4), if I used $A(a, \lambda)A(a, \lambda)=1$, I would get Bell's inequality; just like you and DarMM and many others do using $A(b, \lambda)A(b, \lambda)=1$ in the alternative formulation in Post #51.

But then I would have converted (5) -- by an improper separation of terms* -- from an inequality that is never false into Bell's inequality which is frequently false.

* By an improper separation I mean this: I would have converted $E(a,b)E(a,c)$ into $E(b,c)$; which is unlikely to be true. That a function $F(b,c)$ should equal a function $F(a,b)F(a,c)$ over all $a,b,c$.

I'm genuinely having a hard time understanding this. Where have you shown that $E(a,b)E(a,c)$ can be converted into $E(b,c)$? I'm not following.

Also how do you get from your equation (1) to your equation (2)? Your equations are not mine or those given by Bell. Where are you getting them from?

 

[N88]

I'm genuinely having a hard time understanding this. Where have you shown that $E(a,b)E(a,c)$ can be converted into $E(b,c)$? I'm not following.

In (4), if we let $\{A(a, \lambda)A(b, \lambda)\}\{A(a, \lambda)A(c, \lambda)\}$ reduce to $A(b, \lambda)A(c, \lambda)$ by allowing $A(a, \lambda)A(a, \lambda)=1,$ (as stated) then the integral gives $-E(b,c)$. It is similar to the way you get the same final result.

You and I and Bell start and finish at the same point IF we allow that reduction. Noting that such a reduction moves us from a totally valid inequality to Bell's partially valid inequality.

Also how do you get from your equation (1) to your equation (2)? Your equations are not mine or those given by Bell. Where are you getting them from?

Eqn (1) = eqn (2) by algebra, by way of $\{A(a, \lambda)A(b, \lambda)\}\{A(a, \lambda)A(b, \lambda)\} = 1.$

Peter stressed that Bell's inequality is mathematical. So I am trying to understand how the mathematics goes from a physically valid start to a result that is false under QM.

 

[PeterDonis]

I am trying to understand how the mathematics goes from a physically valid start

You continue to miss the point: since the whole derivation is just a mathematical proof, if the conclusion doesn't match QM, the starting point doesn't either. In other words, what Bell is showing is that his starting point is not physically valid. It looks like it ought to be physically valid, but it isn't.

Since you appear unable to grasp this simple point despite numerous attempts to explain it, there is no point in continuing this thread. Thread closed.