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I am having trouble following the Peskin and Schroeder and their derivations to show that a Dirac particle is a spin 1/2 particle (page 60 and 61). I understand how he gets the first (unnumbered) equation on page 61. However, I don't understand how he gets to the second equation:

\begin{equation}

J_z a^{s \dagger}_0|0 \rangle = \frac{1}{2m} \sum_r \left(u^{s \dagger}(0) \frac{\Sigma^3}{2} u^r(0) \right) a_0^{r \dagger} |0 \rangle

\end{equation}

In particular I do not understand why there is only one "summation" symbol. I would have thought that the equation would be:

\begin{equation}

J_z a^{s \dagger}_0|0 \rangle = \frac{1}{2m} \sum_r \sum_{s} \left(u^{s \dagger}(0) \frac{\Sigma^3}{2} u^r(0) \right) a_0^{r \dagger} |0 \rangle

\end{equation}

Does anybody know what they have done with the second "summation" symbol?

Any help would much appreciated.

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# Proof that a Dirac particle has spin 1/2?

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