Jordan-Schwinger representation

$$\begin{array}{rrl}\widehat{J_x}=& \frac{1}{2}& ({\hat{a}}^{†}\hat{b}+{\hat{a}}^{†}\hat{b})\\ \widehat{J_y}=& -\frac{i}{2}& ({\hat{a}}^{†}\hat{b}-{\hat{a}}^{†}\hat{b})\\ \widehat{J_z}=& \frac{1}{2}& ({\hat{a}}^{†}\hat{a}-{\hat{b}}^{†}\hat{b})\\ {\hat{J}}^2=& & {\widehat{J_x}}^2+{\widehat{J_y}}^2+{\widehat{J_z}}^2\end{array}$$

Where $\hat{a}$, $\hat{b}$, ${\hat{a}}^{†}$ and ${\hat{b}}^{†}$ are the standard creation operators and annihilation operators.