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A Top-Down Account of Linear Canonical Transforms
We contend that what are called Linear Canonical Transforms (LCTs) should be
seen as a part of the theory of unitary irreducible representations of the
'2+1' Lorentz group. The integral kernel representation found by Collins,
Moshinsky and Quesne, and the radial and hyperbolic LCTs introduced thereafter,
belong to the discrete and continuous representation series of the Lorentz
group in its parabolic subgroup reduction. The reduction by the elliptic and
hyperbolic subgroups can also be considered to yield LCTs that act on
functions, discrete or continuous in other Hilbert spaces. We gather the
summation and integration kernels reported by Basu and Wolf when studiying all
discrete, continuous, and mixed representations of the linear group of real matrices. We add some comments on why all should be considered
canonical
The Fourier U(2) Group and Separation of Discrete Variables
The linear canonical transformations of geometric optics on two-dimensional
screens form the group , whose maximal compact subgroup is the Fourier
group ; 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 . Two distinct
subalgebra chains are used to model arrays of points placed along
Cartesian or polar (radius and angle) coordinates, thus realizing one case of
separation in two discrete coordinates. The -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
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