Spin-1 System

This spin is for 3-dimensional Hilbert Space. W/Z bosons, photons and gluons have spin-1. For spin-1 systems,

S_x\triangleq\frac{\hbar}{2}\Sigma_x\quad S_y\triangleq\frac{\hbar}{2}\Sigma_y\quad S_z\triangleq\frac{\hbar}{2}\Sigma_z

i.e.,

S_x\triangleq\frac{\hbar}{\sqrt{2}}\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{pmatrix}\quad S_y\triangleq\frac{\hbar}{\sqrt{2}}\begin{pmatrix}0 & -i & 0\\ i & 0 & -i\\ 0 & i & 0\end{pmatrix}\quad S_z\triangleq\hbar\begin{pmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -1\end{pmatrix}

Note that

\left\lvert 1 \right\rangle=\begin{pmatrix}1 \\ 0 \\ 0\end{pmatrix}\quad \left\lvert 0 \right\rangle=\begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}\quad \left\lvert -1 \right\rangle=\begin{pmatrix}0 \\ 0 \\ -1\end{pmatrix}\quad

For photons, $S_z$ is often called rectilinear and $S_x$ is often called diagonal and $S_y$ are often called circular.