# Pauli algebra

The -dimensional real Clifford algebra generated by the Pauli matrices [a1]

where is the complex unit . The matrices , and satisfy and the anti-commutative relations:

These matrices are used to describe angular momentum, spin- 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 satisfying

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

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 |

**How to Cite This Entry:**

Pauli algebra. G.P. Wene (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Pauli_algebra&oldid=14443