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

  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},
  • K. WolfL. 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… 

Figures from this paper

Finite optical Hamiltonian systems

In this essay we finitely quantize the Hamiltonian system of geometric optics to a finite system that is also Hamiltonian, but where signals are described by complex N-vectors, which are subject to

The Fourier U(2)F group on square and round pixellated arrays

The Fourier group U(2)F is the maximal compact group of the group of linear canonical transformations on configuration space. When we finitely quantize the two-dimensional harmonic oscillator we turn

Unitary rotation of pixellated polychromatic images

Unitary rotations of polychromatic images on finite two-dimensional pixellated screens provide invertibility, group composition, and thus conservation of information. Rotations have been applied on

Goryachev-Chaplygin, Kovalevskaya, and Brdička-Eardley-Nappi-Witten pp-waves spacetimes with higher rank Stäckel-Killing tensors

Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdicka-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher



Finite two-dimensional oscillator: II. The radial model

A finite two-dimensional radial oscillator of (N + 1)2 points is proposed, with the dynamical Lie algebra so(4) = su(2)x⊕su(2)y examined in part I of this work, but reduced by a subalgebra chain

Finite two-dimensional oscillator: I. The Cartesian model

A finite two-dimensional oscillator is built as the direct product of two finite one-dimensional oscillators, using the dynamical Lie algebra su(2)x⊕su(2)y. The position space in this model is a

Discrete Systems and Signals on Phase Space

The analysis of discrete signals —in particular finite N -point signals— is done in terms of the eigenstates of discrete Hamiltonian systems, which are built in the context of Lie algebras and

Fractional Fourier transforms in two dimensions.

  • R. SimonK. Wolf
  • Mathematics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2000
We analyze the fractionalization of the Fourier transform (FT), starting from the minimal premise that repeated application of the fractional Fourier transform (FrFT) a sufficient number of times

Linear Canonical Transformations and Their Unitary Representations

We show that the group of linear canonical transformations in a 2N‐dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N

Rotation and gyration of finite two-dimensional modes.

  • K. WolfT. Alieva
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2008
In finite systems, where the emitters and the sensors are in NxN square pixellated arrays, one defines corresponding finite orthonormal and complete sets of two-dimensional Kravchuk modes through the importation of symmetry from the continuous case, the transformations of the Fourier group are applied on the finite modes.

Canonical Transformations and Matrix Elements

We use the ideas on linear canonical transformations developed previously to calculate the matrix elements of the multipole operators between single‐particle states in a three‐dimensional oscillator

Analysis of digital images into energy-angular momentum modes.

  • L. E. VicentK. Wolf
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2011
The properly orthonormal "Laguerre-Kravchuk" discrete functions Λ(n, m)(q(x), q(y)) are proposed as a convenient basis to analyze the sampled beams into their E-AM polar modes, and with them synthesize the input image exactly.

Unitary rotation of square-pixellated images

Fractional Fourier-Kravchuk transform

We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of