Pauli algebra

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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.


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098