Pauli algebra

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


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



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.


[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
How to Cite This Entry:
Pauli algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by G.P. Wene (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article