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I am reading John Bell's article On the Einstein Podolsky Rosen paradox and I am confused by the following sentence in the beginning of part III:

Suppose we have a spin half particle in a pure spin state with polarization denoted by a unit vector ...

What does Bell mean by polarization of spin $\tfrac12$ state?

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  • $\begingroup$ Spin has a direction. If spin is polarized it just means the spin points along a particular direction. $\endgroup$
    – joseph h
    Commented Jun 4, 2023 at 5:43
  • $\begingroup$ @josephh how spin can be not polarized? $\endgroup$
    – kludg
    Commented Jun 4, 2023 at 7:00
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    $\begingroup$ Because you need a direction along with the spin orientation. You cannot say that because a particle has spin direction, then it's automatically polarized. In this instance, the particle spin is polarized along the direction denoted by the unit vector. $\endgroup$
    – joseph h
    Commented Jun 4, 2023 at 7:06
  • $\begingroup$ @josephh Well, you confused things completely. What is spin orientation? Spin can be measured along any orientation. How it differs from spin polarization? $\endgroup$
    – kludg
    Commented Jun 4, 2023 at 7:23
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    $\begingroup$ The particles are prepared in such a way that a spin measurement will always reveal a spin parallel to the unit vector. This is called "polarization." $\endgroup$
    – tobi_s
    Commented Jun 5, 2023 at 2:42

2 Answers 2

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The space of states of a spin-1/2 object is the Bloch sphere, which, as the name implies, can be viewed as the two-sphere $S^2\subset\mathbb{R}^3$. Every point of the sphere corresponds to a unit vector $\hat{n}\in\mathbb{R}^3$, and the state corresponding to it is an eigenstate of the spin operator $S_{\hat{n}}$ that measures the spin in that direction. This $\hat{n}$ is the "polarization vector".

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  • $\begingroup$ I guess you are correct, there are no other directions in 3D for spin-1/2 than directions on the Bloch sphere. I'm still confused what is spin polarization for other spin values but I did not ask it. Hopefully your answer pushed me in right direction. $\endgroup$
    – kludg
    Commented Jun 5, 2023 at 7:29
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A spin 1/2 state can be projected on an axis as +1/2 or -1/2, it is an angular momentum vector with its units. Bell wants a unit-less vector, to denote the direction of spin as + or -1, and calls it the polarization vector .

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  • $\begingroup$ I've no problems with 1 vs 1/2: Bell uses Pauli matrices instead of spin operators. What confuses me is the word polarization. I guess it is a direction where a particle has spin eigenvalues, that is spin of a particle is predetermined. $\endgroup$
    – kludg
    Commented Jun 4, 2023 at 6:04
  • $\begingroup$ the direction of the spin is given by the polarization vector. $\endgroup$
    – anna v
    Commented Jun 4, 2023 at 8:33
  • $\begingroup$ Sorry, but I have no idea what is spin direction or spin polarization vector. To my knowledge, spin can be measured in any direction. $\endgroup$
    – kludg
    Commented Jun 4, 2023 at 8:52
  • $\begingroup$ Spin is angular momentum and it adds up to the angular momentum of a system. Angular momentum is a vector, Vectors britannica.com/science/vector-physics $\endgroup$
    – anna v
    Commented Jun 4, 2023 at 18:41

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