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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.
pdf (1792 kb)
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