Why Does Bell's Calculation Yield a Negative Cosine?

[Adam Lewis]

Hello,

I am trying to reproduce Bell's calculation for the expectation value of paired spin measurements on particles in the singlet state. For unit vectors $\hat{a}$ and $\hat{b}$ we want to calculate

$$P(a,b)=<\psi|(\hat{a}\cdot\vec{\sigma})(\hat{b} \cdot \vec{\sigma})|\psi>$$

where $|\psi>$ is the singlet state.

Via the commutation and anticommutation relations for the Pauli matrices the enclosed operator is simply

$$(\hat{a}\cdot\hat{b})I + \imath\vec{\sigma}\cdot(\hat{a}\times\hat{b}).$$

As a scalar the dot product can be pulled from the bra-ket, leaving $(\hat{a}\cdot\hat{b})<\psi|I|\psi>=(\hat{a}\cdot \hat{b})$ since the singlet state is normalized. The cross product's expectation value turns out to vanish. Thus the final answer is

$$P(a,b)=(\hat{a}\cdot\hat{b})=\cos(\theta).$$

The answer usually quoted, however, is $-\cos(\theta)$, and I can't figure out where the minus sign is coming from. Any ideas?

 

[Adam Lewis]

Well, that would get you the minus sign. But I had thought the fact the spins were anti-parallel to be already encoded by the singlet state. It seems odd to me that you should have to insert this information again via the operator. Maybe I'm misunderstanding how $\vec{\sigma}$ is supposed to work?