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

摘要

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