Show that the Heisenberg spin operator S can be written in second quantized form as

\begin{eqnarray*}\nonumber
S_{x}& = &\phantom{+} \frac{1}{2}(c _{\uparrow}^{\dag...
...arrow} - c_{\downarrow}^{ \dagger} c_{\downarrow}). \\ \nonumber
\end{eqnarray*}



Check the commutation relations!