# A geometric algebra reformulation of 2x2 matrices: the dihedral group D 4 in bra-ket notation - Mathematical Physics

A geometric algebra reformulation of 2x2 matrices: the dihedral group D 4 in bra-ket notation - Mathematical Physics - Download this document for free, or read online. Document in PDF available to download.

Abstract: 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.

Author: ** Quirino M. Sugon Jr., Carlo B. Fernandez, Daniel J. McNamara**

Source: https://arxiv.org/