A pion decays to a singlet electron/positron state. We will measure the component of the electron's spin in the $\vec a$ direction and positron's spin in the $\vec b$ direction. If there exists a local hidden variable(s) $\lambda$, then outcomes of the measurements must be given by functions $A(\vec a,\lambda)$ and $B(\vec b,\lambda)$ respectively. these functions can only output the values $\pm1$ (in units of $\hbar/2$.) Due to entanglement and the case of $\vec a=\vec b$, we may write $B(\vec b,\lambda)=-A(\vec b,\lambda)$. We also assume a non-negative probability density $\rho(\lambda)$ which integrates to unity over all of space. Then Bell's inequality follows in the usual way.
This is my question: What is it in the above definitions for $\lambda$ that makes it "local"?
Griffiths writes the following:
Is the locality condition only that $A$ is strictly not a function of $\vec b$ (and $B$ not of $\vec a$), which could not be ruled out for FTL signals?
Best Answer
In addition to $A$ strictly not being a function of $b$ (and vice versa), the locality condition is ensures that $\vec{a}$, $\vec{b}$, and $\lambda$ are completely independent of each other. In particular, $\vec{a}$, $\vec{b}$ are not functions of $\lambda$ (or correlated with $\lambda$ in any way). Any such correlation would spoil the proof of Bell's theorem.