A geometric algebra reformulation of 2x2 matrices: the dihedral group D_4 in bra-ket notation

TL;DR

This paper reformulates 2x2 matrices and the dihedral group D_4 using geometric algebra, providing new insights into rotation operators, group algebra, and their relation to Fermion matrices and Pauli spin matrices.

Contribution

It introduces a geometric algebra approach to represent dihedral group D_4 and 2x2 matrices, revealing new geometric interpretations of spin and rotation operators.

Findings

D_4 is represented as a subgroup of SO_3 using Clifford algebra.

Fermion matrices are expressed as linear combinations of D_4 rotation operators.

Pauli matrices are reinterpreted as vector rotation operators in geometric algebra.

Abstract

We represent vector rotation operators in terms of bras or kets of half-angle exponentials in Clifford (geometric) algebra Cl_{3,0}. We show that SO_3 is a rotation group and we define the dihedral group D_4 as its finite subgroup. We use the Euler-Rodrigues formulas to compute the multiplication table of D_4 and derive its group algebra identities. We take the linear combination of rotation operators in D_4 to represent the four Fermion matrices in Sakurai, which in turn we use to decompose any 2x2 matrix. We show that bra and ket operators generate left- and right-acting matrices, respectively. We also show that the Pauli spin matrices are not vectors but vector rotation operators, except for \sigma_2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume.