Question on John Bell's Original Paper

[43arcsec]

Hi Forum, I am trying to follow the math in Bell's original paper, and I am getting tripped up on equation (14). Does anyone know the mathematic legerdemain Bell used to go from the first equation below to the 2nd?

$\begin{align} P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int{d\lambda\rho(\lambda)[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]} \\ =\int{d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]} \end{align}$

Here is a link to the original paper:

http://www.drchinese.com/David/Bell.pdf

I hope I am not missing something trivial, but thanks for you help!

 

[stevendaryl]

Hi Forum, I am trying to follow the math in Bell's original paper, and I am getting tripped up on equation (14). Does anyone know the mathematic legerdemain Bell used to go from the first equation below to the 2nd?

$\begin{align} P(\vec{a},\vec{b})-P(\vec{a},\vec{c})=-\int{d\lambda\rho(\lambda)[A(\vec{a},\lambda)A(\vec{b},\lambda)-A(\vec{a},\lambda)A(\vec{c},\lambda)]} \\ =\int{d\lambda\rho(\lambda)A(\vec{a},\lambda)A(\vec{b},\lambda)[A(\vec{b},\lambda)A(\vec{c},\lambda)-1]} \end{align}$

Here is a link to the original paper:

http://www.drchinese.com/David/Bell.pdf

I hope I am not missing something trivial, but thanks for you help!

What you might be missing is that $A(\vec{b},\lambda) = \pm 1$.

So when you multiply it out:

$A(\vec{a},\lambda) A(\vec{b},\lambda) [A(\vec{b},\lambda) A(\vec{c},\lambda) - 1]$
$= A(\vec{a},\lambda) \underbrace{(A(\vec{b},\lambda) A(\vec{b},\lambda))}_{= 1} A(\vec{c},\lambda) - A(\vec{a},\lambda) A(\vec{b},\lambda)]$
$= A(\vec{a},\lambda) A(\vec{c},\lambda) - A(\vec{a},\lambda) A(\vec{b},\lambda)]$

 

[43arcsec]

Thanks Steve, that was perfect. No chance I was coming up with the multiply by 1 trick, although multiplying by one seems to be a favorite of theorists.

I have another problem with Bell's proof and wonder if you can help. In the derivation of (18) from (16) and (17), apparently the addition of the two, it appears (18) is incorrect:

$$\begin{align} | \bar{P}(\vec{a}\cdot\vec{b})+\vec{a}\cdot\vec{b}| \leq \epsilon \text{ (16)} \\\ | \overline{\vec{a} \cdot \vec{b}}- \vec{a}\cdot\vec{b}| \leq \delta \text{ (17)}\\\ | \bar{P}(\vec{a}\cdot\vec{b})+\vec{a}\cdot\vec{b}| \leq \delta+\epsilon \text{ (18)} \\\\\ \text{ it seems like (18) should be }\\\\\ | \bar{P}(\vec{a}\cdot\vec{b})+\overline{\vec{a} \cdot \vec{b}}| \leq \delta+\epsilon \text{ (revised 18)} \\\ \end{align}$$

I find it hard to believe this is a typo and much easier to believe I am missing something with the inequalities. Thanks again for you help.

 

[Avodyne]

I think it's a typo. There are other, more obvious typos in the paper, so it seems like it wasn't proofed especially well.

But I can't help but wonder why you want to go through Bell's original analysis; much more streamlined (and clear) analyses are available now. See, e.g., "Derivation of CHSH inequality" in http://en.wikipedia.org/wiki/Bell's_theorem. (As explained in "CHSH inequality" in the same wikipedia article, Bell's inequality is a special case of CSHS.)

 

[DrClaude]

I think it's a typo.

I don't think it's a typo, as he uses eq. (18) as is to write eq. (20).

I think the confusion comes from thinking that the LHS of (16) and (17) are added together. My reading is simply that since
$$
| \bar{P}(\vec{a}\cdot\vec{b})+\vec{a}\cdot\vec{b}| \leq \epsilon \quad \text{ (16)}
$$
a weaker inequality can be built using any $\delta \geq 0$, i.e.,
$$
| \bar{P}(\vec{a}\cdot\vec{b})+\vec{a}\cdot\vec{b}| \leq \delta+\epsilon \quad \text{ (18)}
$$
It just happens that by taking a specific $\delta$,
$$
| \overline{\vec{a} \cdot \vec{b}}- \vec{a}\cdot\vec{b}| \leq \delta \quad \text{ (17)}
$$
he gets the contradiction (22) that is significant.

 

[43arcsec]

Yes, DrClaude, I believe you are correct: the equations are not added. That simple alternative just didn't occur to me.

Thanks for the link Avodyne, I will check it out. You may be right regarding Bell's paper, but in general I have usually found original papers to be an excellent educational source. Einstein's paper on a General Relativity is one of the best examples.

Thanks to both of you for your comments, I greatly appreciate it.

 

[Avodyne]

Bell's paper is most definitely worth reading for what he has to say about the result; it's just that the math details can be significantly simplified, so IMO they're not worth sweating over.