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arXiv · 0811.3680

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

Quirino M. Sugon Jr.·Carlo B. Fernandez·Daniel J. McNamara
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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 σ_2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume.

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Quirino M. Sugon Jr., Carlo B. Fernandez, Daniel J. McNamara. 2008-11-24. A geometric algebra reformulation of 2x2 matrices: the dihedral group D_4 in bra-ket notation. https://arxiv.org/abs/0811.3680

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