Parametrizing Clifford Algebras' Matrix Generators with Euler Angles

TL;DR

This paper introduces a parametrization of Clifford algebra matrix generators using Euler angles, enabling a rotation-based interpretation and facilitating decompositions relevant to group theory and quantum mechanics.

Contribution

It constructs a new Euler angle parametrization of Clifford algebra generators, linking them to well-known matrices and enabling subgroup parametrizations within SU(4).

Findings

Parametric matrix generators are expressed via Kronecker products of Pauli matrices.

Decomposition of generators into Pauli, Dirac, and Gell-Mann matrices established.

A parametrization of SU(4) subgroups using four-vector parameters is achieved.

Abstract

A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd-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 for a given algebra. 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 2nd and 4th-order are linearly decomposed in terms of Pauli, Dirac, and 4th-order Gell-Mann matrices establishing a direct correspondence between the 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 4th-order matrix generators and in terms of four-vector parameters is explored. By establishing unitary conditions, a parametrization of two sub-groups of SU(4) is achieved.