Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 053, 18 pages      arXiv:1106.0093      https://doi.org/10.3842/SIGMA.2011.053
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

The Fourier U(2) Group and Separation of Discrete Variables

Kurt Bernardo Wolf a and Luis Edgar Vicent b
a) Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Mor. 62210, México
b) Deceased

Received February 19, 2011, in final form May 26, 2011; Published online June 01, 2011

Abstract
The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,R), whose maximal compact subgroup is the Fourier group U(2)F; this includes isotropic and anisotropic Fourier transforms, screen rotations and gyrations in the phase space of ray positions and optical momenta. Deforming classical optics into a Hamiltonian system whose positions and momenta range over a finite set of values, leads us to the finite oscillator model, which is ruled by the Lie algebra so(4). Two distinct subalgebra chains are used to model arrays of N2 points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N2-vectors in this space are digital (pixellated) images on either of these two grids, related by a unitary transformation. Here we examine the unitary action of the analogue Fourier group on such images, whose rotations are particularly visible.

Key words: discrete coordinates; Fourier U(2) group; Cartesian pixellation; polar pixellation.

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