Numerical Calculations in Clifford Algebra

8Pauli Matrices

Pauli (1927) matrices were constructed in the context of quantum mechanics involving electrons.

8.1 Theory

Pauli matrices are represented in Clifford algebra by identifying the unit Pauli matrices¹ with particular Clifford units:

(8.1)

in a context, where the signature is all . Pauli matrices form a particular subset of three‐dimensional Clifford numbers. Although three‐dimensional Clifford numbers have four grades and eight components, Pauli matrices retain only two of the grades (zero and two) and four of the components.

The unit Pauli matrices obey the rules for multiplication:

(8.2)

These rules are honoured by the Clifford algebra when Grassmann's central multiplication is used.

Comparison of Eq. (8.1) to the equivalent identification for quaternions in Eq. (7.1) indicates that the Pauli matrices and quaternions are closely related. The non‐scalar Pauli units ...