A (probably) simple question about Bell's paper

  • #1

Summary:

Specifically, the demonstration of a single-particle hidden variable
Hello! I am trying to understand Bell's Inequality and although I can follow the arguments of the inequality as are mentioned in modern texts, something was always bothering me. In section III (Illustration) of the original paper, equation (5) states:

< σ⋅a > = 1 - 2θ'/π​

But for the life of me, I don't know why, and I am not sure how to assemble this by hand, it seems like everywhere I find similar values there's just a "well, obviously this gives: " followed by the above. Trying to put together the integral has me wondering if B(b,λ) is just to be taken as 1? Maybe it's been too long since I did any calculus...
 

Answers and Replies

  • #2
wle
316
139
You don't need to do any calculus. You're averaging ##\text{sign} \, \boldsymbol{\lambda} \cdot \boldsymbol{a}'## over vectors ##\boldsymbol{\lambda}## that satisfy ##\boldsymbol{\lambda} \cdot \boldsymbol{p} > 0## for some given vectors ##\boldsymbol{a}'## and ##\boldsymbol{p}##. ##\text{sign} \, \boldsymbol{\lambda} \cdot \boldsymbol{a}'## is ##1## if ##\boldsymbol{\lambda} \cdot \boldsymbol{a}' > 0## and ##-1## if ##\boldsymbol{\lambda} \cdot \boldsymbol{a}' < 0##. So the question is: out of all ##\boldsymbol{\lambda}##s in the hemisphere ##\boldsymbol{\lambda} \cdot \boldsymbol{p} > 0## what fraction are also in the hemisphere ##\boldsymbol{\lambda} \cdot \boldsymbol{a}' > 0## and what fraction are in the hemisphere ##\boldsymbol{\lambda} \cdot \boldsymbol{a}' < 0##?
 
  • #3
Of course, within minutes of explaining the problem in this post I managed to figure it out randomly while traveling to the pub, imagine my surprise when I come back here to find that you told me the same thing. Thank you for confirming what I suspected - I appreciate you helping me do a sanity check!

Edit: I suppose I should find a way to change my username to "considerablylessconfused"
 

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