Revisiting 2×2 matrix optics: Complex vectors, Fermion combinatorics, and Lagrange invariants

Quirino M. Sugon Jr. and Daniel J. McNamara

(Submitted on 3 Dec 2008)

Abstract: We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray’s 2×2 matrix operators should be right-acting, so that the matrix product succession would go with light’s left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants.

Comments: 10 pages, 6 figures
Subjects: Optics (physics.optics)
Cite as: arXiv:0812.0664v1 [physics.optics]

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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)