Physical significance of a.σ in expectation -E(a.σ b.σ)?

[PeterDonis]

The relevant point about the model of @N88 is that it is exactly the type of model that Bell was proving his theorem about.

I agree, but I'm not sure @N88 does. But whether he does or not, the discussion here is about Bell's paper, so models that are not the type that Bell was talking about in his paper are off topic.

 

[stevendaryl]

I agree, but I'm not sure @N88 does. But whether he does or not, the discussion here is about Bell's paper, so models that are not the type that Bell was talking about in his paper are off topic.

Right.

 

[Jilang]

Thank you Steven Daryl for post #69. I have been drawing out the orange segments and I have it now!

 

[N88]

No, $\hat{a}$ is a unit vector pointing in a particular direction. The result of this dot product is an operator that represents measuring spin about the axis $\hat{a}$.

These look ok to me.

I don't know about these, since no unit vector $\hat{a}$ representing the direction of the spin axis to be measured is included.

Thank you. In terms of the OP, how's this:

$(\hat{a}\cdot\boldsymbol{\sigma}_{1})$ represents the 'dot-product' of a unit-vector $\hat{a}$ with $\boldsymbol{\sigma}_{1}$, the Pauli 'vector', a vector of Pauli matrices.

$\boldsymbol{\sigma}_{1}=\{{\sigma}_{x1},{\sigma}_{y1},{\sigma}_{z1}\}$. (1)

To be clear to beginners (like myself), I call it a 'dot-product' because, although $\boldsymbol{\sigma}_{1}$ is not a normal vector, the product yields:

$\hat{a}\cdot\boldsymbol{\sigma}_{1}$ = ${a}_x{\sigma}_{x1}+{a}_y{\sigma}_{y1}+{a}_z{\sigma}_{z1}$. (2)

[Peter, re this from you: "The result of this 'dot-product' is an operator that represents measuring spin about the axis $\hat{a}$." Is that still the meaning of (2)?]

Outcomes include the expectations: $\left\langle \hat{a}\cdot\boldsymbol{\sigma}_{1}\right\rangle =0; (3) \left\langle (\hat{a}\cdot\boldsymbol{\sigma}_{1})^{2}\right\rangle =1$. (4)

$\left\langle{\sigma}_{x1}^{2}\right\rangle=\left\langle{\sigma}_{y1}^{2}\right\rangle=\left\langle{\sigma}_{z1}^{2}\right\rangle=\left\langle{\sigma}_{1}^{2}\right\rangle/3=1$. (5)

Thanks.

Well, I interpreted $\sigma_{x1}$ as the operator $\hat{a} \cdot \vec{\sigma_1}$ in the case $\hat{a} = \hat{x}$.

Does this fit with what I've summarised above? Thanks

 

[PeterDonis]

Peter, re this from you: "The result of this 'dot-product' is an operator that represents measuring spin about the axis $\hat{a}$." Is that still the meaning of (2)?

Yes.

 

[N88]

If $\circ$ is a function (that is, $\vec{a} \circ \vec{\lambda}$ always gives the same result for the same values of $\vec{a}$ and $\vec{\lambda}$), then your model satisfies CFD: If Alice had chosen $\vec{a'}$ instead of $\vec{a}$, her result would have definitely been $\vec{a'} \circ \vec{\lambda}$.

$\circ$ is a function: it always gives the same result for the same $\vec{\lambda}$. IF Alice had measured under the detector-orientation $\vec{a'}$ instead of $\vec{a}$, THEN her result would have definitely been $\vec{a'} \circ \vec{\lambda}$.

Please detail your definition of CFD in this situation. Thanks.

 

[PeterDonis]

Please detail your definition of CFD in this situation.

He already did, in the very statement you quoted.

As I've already said, further discussion of your personal model is off topic.

 

[N88]

He already did, in the very statement you quoted.

As I've already said, further discussion of your personal model is off topic.

Can I discuss CFD here?

 

[PeterDonis]

Can I discuss CFD here?

In connection with the model described in Bell's paper, yes.

 

[stevendaryl]

$\circ$ is a function: it always gives the same result for the same $\vec{\lambda}$. IF Alice had measured under the detector-orientation $\vec{a'}$ instead of $\vec{a}$, THEN her result would have definitely been $\vec{a'} \circ \vec{\lambda}$.

Please detail your definition of CFD in this situation. Thanks.

That is the definition of CFD. A model satisfies CFD if it gives definite answers to questions of the type: "If the experimenter had done A instead of B, her result would have been X instead of Y."

 

[N88]

That is the definition of CFD. A model satisfies CFD if it gives definite answers to questions of the type: "If the experimenter had done A instead of B, her result would have been X instead of Y."

This is rushed but it should get us to a point that we can progress from.

OK; thank you. But that is a different form of CFD to the one Bell proceeds under.*

Note that my CFD is licensed under EPRB by Bell (1964), eqns (1) and (13). Which means this: I can convert my CFD setting to an experimental test and it passes adequately. So mine is that special form of CFD that can be adequately proven; in this case consistent with Bell's model.

*A second variant of CFD can be adequately disproven. Bell demonstrates that when he derives his inequalities under EPRB and they are experimentally refuted. I do not use this variant and I am not conflicted by AAD and nonlocality: whereas, in his final year (1990), Bell talked about his dilemma on this front. Further, he was confident that someone would come up with answer; even that it might show that he had been rather silly.**
** http://www.quantumphil.org./Bell-indeterminism-and-nonlocality.pdf

Further, I share his motivation: "It is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called ‘hidden variable' possibility,” Bell (Speakable and Unspeakable, 2004:28-29). So, all in all, I'm a keen student of Bell and keen to learn the more about the QM aspects of his search.

Therefore: I have no wish to breach PF policies about private studies; etc. But in answering past questions I had need to refer to some components of it; all based on Bell's model but not widely accepted.

So, as to the way ahead: I suggest that I only respond to specific questions and do not move beyond them; ie, I let comments like this slide by: So [N88's] details are completely unimportant, since Bell proved a fact about all such models.

Excuse rush here; and I do appreciate and welcome our exchanges; I learn much from such. Hope this is OK.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

my CFD

Is off topic here. The term "counterfactual definiteness" does not appear in Bell's paper at all, and Bell's paper is the topic of discussion. At this point we have wandered very far from the OP of this thread, and the question in the OP has been thoroughly discussed. The thread will remain closed.