Parametrizing Clifford Algebras’ Matrix Generators with Euler Angles

Abstract

A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd non-degenerate Clifford algebras is constructed by means of the Kronecker product of a parametrized version of Pauli matrices and by the identification of all possible anticommutation sets. The internal parametrization of the matrix generators allows a straightforward interpretation in terms of rotations, and in the absence of a similarity transformation can be reduced to the canonical representations by an appropriate choice of parameters. The parametric matrix generators of second and fourth-order are linearly decomposed in terms of Pauli, Dirac, and fourth-order Gell–Mann matrices establishing a direct correspondence between the different representations and matrix algebra bases. In addition, and with the expectation for further applications in group theory, a linear decomposition of GL(4) matrices on the basis of the parametric fourth-order matrix generators and in terms of four-vector parameters is explored. By establishing unitary conditions, a parametrization of two subgroups of SU(4) is achieved.