Pauli algebra

The $2^3$-dimensional real Clifford algebra generated by the Pauli matrices [a1]

\begin{equation}\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\end{equation}

where $i$ is the complex unit $\sqrt{-1}$. The matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$ satisfy $\sigma^2_x=\sigma^2_y=\sigma^2_z=1$ and the anti-commutative relations:

\begin{equation}\sigma_i\sigma_j+\sigma_j\sigma_i=0\text{ for }i,j\in\{x,y,z\}.\end{equation}

These matrices are used to describe angular momentum, spin-$1/2$ fermions (which include the electron) and to describe isospin for the neutron, proton, mesons and other particles.

The angular momentum algebra is generated by elements $\{J_1,J_2,J_3\}$ satisfying

\begin{equation}J_1J_2=J_2J_1=iJ_3\end{equation}

\begin{equation}J_2J_3-J_3J_2=iJ_1J_3J_1-J_1J_3=iJ_2.\end{equation}

The Pauli matrices provide a non-trivial representation of the generators of this algebra. The correspondence

\begin{equation}1\leftrightarrow\begin{pmatrix}1&0\\0&1\end{pmatrix},I\leftrightarrow i\sigma_1,J\leftrightarrow i\sigma_2,K\leftrightarrow i\sigma_3\end{equation}

leads to a realization of the quaternion division algebra (cf. also Quaternion) as a subring of the Pauli algebra. See [a2], [a3] for algebras with three anti-commuting elements.

References

[a1]	W. Pauli, "Zur Quantenmechanik des magnetischen Elektrons" Z. f. Phys. , 43 (1927) pp. 601–623
[a2]	Y. Ilamed, N. Salingaros, "Algebras with three anticommuting emements I: spinors and quaternions" J. Math. Phys. , 22 (1981) pp. 2091–2095
[a3]	N. Salingaros, "Algebras with three anticommuting elements II" J. Math. Phys. , 22 (1881) pp. 2096–2100