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Journal of Physics Conference Series

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

Kurt Bernardo Wolf, Natig M. Atakishiyev

Published Feb 8, 2012• 0 citations
Publisher

Abstract

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 this space into an array of N 2 points arranged along either Cartesian or polar coordinates, as a square or a circle, that model pixellated screens. We investigate the action of U(2) F on these two N 2 -dimensional spaces, which become a homogeneous space for this group action.

DOI
10.1088/1742-6596/343/1/012130

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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 this space into an array of N 2 points arranged along either Cartesian or polar coordinates, as a square or a circle, that model pixellated screens. We investigate the action of U(2) F on these two N 2 -dimensional spaces, which become a homogeneous space for this group action.
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 square grid of points. While the ordinary `continuous' two-dimensional quantum oscillator has a symmetry algebra u(2), the symmetry algebra of the finite model is only u(1)x⊕u(1)y, because it lacks rotations in the position (and momentum) plane. We show how to `import' an SO(2) group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid. We thus propose a finite analogue for fractional U(2) Fourier transforms.
A unitary transformation between Cartesian and polar pixellations of finite two-dimensional images is obtained from the su(2) model for discrete and finite signals. This transformation analyzes the original image into its finite Cartesian "Laguerre-Kravchuk" modes (involving Wigner little-d functions) and synthesizes it back using a polar mode basis with the same set of mode coefficients. The polar basis is derived from the quantum angular momentum theory, and its modes are given by Clebsch-Gordan coefficients.
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 coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.
Quantum mechanics of a linear harmonic oscillator in a finite-dimensional Hilbert space satisfying the correct equations of motion is studied. The connections to Weyl's formulation of the algebra of bounded unitary operators in finite space as well as to a truncated quantized linear harmonic oscillator are discussed. It is pointed out that the discrete Fourier transformation (DFT) plays a central role in determining the actual form of the position, the momentum, the number and the Hamiltonian operators. The explicit form of these operators in different bases is exhibited for some low values of the dimension of the Hilbert space. In this formulation, it is shown that the Hamiltonian is indeed the logarithm of the DFT and that by modifying Weyl's framework to include position and momentum operators with non-uniformly spaced spectra the equations of motion are satisfied.
Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su(333) algebra of Cartesian three-dimensional oscillator multiplet states into so(333) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.