• DOI:10.3842/SIGMA.2011.053
• Corpus ID: 18682267

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

@article{Wolf2011TheFU,
  title={The Fourier U(2) Group and Separation of Discrete Variables},
  author={Kurt Bernardo Wolf and Luis Edgar Vicent},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2011},
  volume={7},
  pages={053}
}

• K. Wolf, L. E. Vicent
• Published 1 June 2011
• Mathematics
• Symmetry Integrability and Geometry-methods and Applications

The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4;<), whose maximal compact subgroup is the Fourier group U(2) F ; this includes isotropic and anisotropic Fourier transforms, screen rotations and gy- rations 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 dis- tinct subalgebra chains are used to model arrays of N 2 points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N 2 -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.