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

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

(Submitted on 24 Nov 2008)

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 2×2 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.

Comments: 11 pages, 3 figures, 1 table
Subjects: Mathematical Physics (math-ph)
Cite as: arXiv:0811.3680v1 [math-ph]
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Submission history
From: Quirino Sugon Jr. [view email]
[v1] Mon, 24 Nov 2008 04:38:07 GMT (17kb)

Taxonomy of Clifford Cl_{3,0} subgroups: Choir and band groups

Quirino M. Sugon Jr., Daniel J. McNamara

(Submitted on 2 Sep 2008)

Abstract: We list the subgroups of the basis set of Cl_{3,0} and classify them according to three criteria for construction of universal Clifford algebras: (1) each generator squares to +1 or -1, (2) the generators within the group anticommute, and (3) the order of the resulting group is 2^{n+1}, where n is the number of nontrivial generators. Obedient groups we call choirs; disobedient groups, bands. We classify choirs by modes and bands by rhythms, based on canonical equality. Each band generator has a transposition (number of other generators it commutes with). The band’s transposition signature is the band’s chord. The sum of transpositions divided by twice the number of generator pair combinations is the band’s beat. The band’s order deviation is the band’s disorder. For n less than or equal 3, we show that the Cl_{3,0} basis set has 21 non-isomorphic subgroups consisting of 9 choirs and 12 bands.

Comments: 9 pages, 10 tables
Subjects: Mathematical Physics (math-ph)
Cite as: arXiv:0809.0351v1 [math-ph]
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Submission history
From: Quirino Sugon Jr. [view email]
[v1] Tue, 2 Sep 2008 05:41:24 GMT (13kb)