Pauli algebra

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

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$$

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

$${\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i=0\text{ for }i,j\in \{x,y,z\}.$$

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

$$J_1{J}_2=J_2{J}_1=i{J}_3$$

$$J_2{J}_3-J_3{J}_2=i{J}_1{J}_3{J}_1-J_1{J}_3=i{J}_2.$$

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

$$1\leftrightarrow \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),I\leftrightarrow i{\sigma }_1,J\leftrightarrow i{\sigma }_2,K\leftrightarrow i{\sigma }_3$$

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.

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