# Quantum Mechanics - Hidden Variables

In Steven Weinberg's Lecture on Quantum Mechanics (p. 342), he writes:

The correlation between the spins of the two particles can be expressed as the average value of the product of the $\hat{a}$ component of the spin of particle 1 and the $\hat{b}$ component of the spin of particle 2:

$\langle (\textbf{s1} \cdot \hat{a}) (\textbf{s2} \cdot \hat{b}) \rangle = -\frac{\hbar^2}{4} \int{d\lambda\ \rho(\lambda) S(\hat{a}, \lambda) S(\hat{b}, \lambda)}$

where $\hat{a}$ and $\hat{b}$ are any two unit vectors.

In quantum mechanics, we have:

$\langle (\textbf{s1} \cdot \hat{a}) (\textbf{s2} \cdot \hat{b}) \rangle = -\frac{\hbar^2}{4} \hat{a} \cdot \hat{b}$

There is no obstacle to constructing a function $S$ and a probability density $\rho$ for which are equal for any single pair of directions $\hat{a}$ and $\hat{b}$.

Can someone explain to me precisely how this is done and what $S(\hat{a}, \lambda)$ and $\rho(\lambda)$, along with the hidden variables $\lambda$ would be?

My guess is that since we want something that is rotationally invariant since it depends only on the dot product, we want some $S$ and hidden variables $\lambda$ that act nicely under rotations.

On the other hand, since the first equation is a local hidden variable theory, I thought this was not possible by Bell's inequalities...

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I edited it to try and make it clear which parts are quoted from Weinberg and which parts are your own text - but I'm not 100% sure I got it right - can you check? – Nathaniel Apr 22 '13 at 9:23
Yes, this is correct. Thank you so much! – bob riley Apr 22 '13 at 9:25