A geometric algebra reformulation of 2x2 matrices: the dihedral group D4 in bra-ket notation
Abstract
We represent vector rotation operators in terms of bras or kets of half-angle exponentials in Clifford (geometric) algebra Cl3,0. We show that SO3 is a rotation group and we define the dihedral group D4 as its finite subgroup. We use the Euler-Rodrigues formulas to compute the multiplication table of D4 and derive its group algebra identities. We take the linear combination of rotation operators in D4 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 σ2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume.