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)

Electromagnetic energy-momentum equation without tensors: a geometric algebra approach

Quirino M. Sugon Jr., Daniel J. McNamara

(Submitted on 9 Jul 2008)

Abstract: In this paper, we define energy-momentum density as a product of the complex vector electromagnetic field and its complex conjugate. We derive an equation for the spacetime derivative of the energy-momentum density. We show that the scalar and vector parts of this equation are the differential conservation laws for energy and momentum, and the imaginary vector part is a relation for the curl of the Poynting vector. We can show that the spacetime derivative of this energy-momentum equation is a wave equation. Our formalism is Dirac-Pauli-Hestenes algebra in the framework of Clifford (Geometric) algebra Cl_{4,0}.

Comments: 5 pages, no figures.
Subjects: Classical Physics (physics.class-ph)
Cite as: arXiv:0807.1382v1 [physics.class-ph]
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