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13 pages, 327 KiB

Open AccessArticle

2 × 2 Matrices: Manifolds, Realizations, Applications

by Kenan Uriostegui and Kurt Bernardo Wolf

Appl. Sci. 2021, 11(16), 7479; https://doi.org/10.3390/app11167479 - 14 Aug 2021

Cited by 2 | Viewed by 1949

Both geometric and wave optical models, as well as classical and quantum mechanics, realize linear transformations with matrices; for plane optics, these are

2 \times 2

and of unit determinant. Students and some researchers could assume that the structure of this matrix group [...] Read more.

Both geometric and wave optical models, as well as classical and quantum mechanics, realize linear transformations with matrices; for plane optics, these are

2 \times 2

and of unit determinant. Students and some researchers could assume that the structure of this matrix group is fairly evident and hardly interesting. However, the properties and applications even of this lowest

2 \times 2

case are already unexpectedly rich. While in mechanics they cover classical angular momentum, quantum spin, and represent ‘

2 + 1

’ relativity, in optical models they lead from the geometrical description of light propagation in the paraxial regime to wave optics via linear canonical transforms requiring a more penetrating view of their manifold structure and multiple covers. The purpose of this review article is to highlight the topological space of

2 \times 2

matrices as it applies to classical versus quantum and wave models, to underline how the latter requires the double cover of the former, thus using

2 \times 2

matrices as an alternative viewpoint of the quantization process, beside the traditional characterization by commutation and non-commutations of position and momentum. Full article

(This article belongs to the Special Issue Geometrical Optics: Theoretical Achievements and Applications)

24 pages, 225 KiB

Open AccessArticle

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

by Natig M. Atakishiyev, Maurice R. Kibler and Kurt Bernardo Wolf

Symmetry 2010, 2(3), 1461-1484; https://doi.org/10.3390/sym2031461 - 12 Jul 2010

Cited by 10 | Viewed by 6945

Abstract

We propose a group-theoretical approach to the generalized oscillator algebra A_κ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for A_κ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices. Full article

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