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Journal of Physics A Mathematical and General

Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach

S Abe, John T. Sheridan

Published Jun 21, 1994• 90 citations
Publisher

Abstract

We develop the operator formalism to show how systematically the fractional Fourier transformation of a wavefunction, recently introduced by Namias (1980), can be derived from the rotation of the corresponding Wigner distribution function in phase space. In this formalism, the phase factor obtained by McBride and Kerr (1987) is seen to come from the caustics of the harmonic oscillator Green function. Then the idea is generalized to the case of an arbitrary area-preserving linear transformation in phase space, and a concept of the special affine Fourier transformation (SAFT) is introduced. An explicit form of the integral representation of the SAFT is given, and some simple examples are presented.

DOI
10.1088/0305-4470/27/12/023

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We develop the operator formalism to show how systematically the fractional Fourier transformation of a wavefunction, recently introduced by Namias (1980), can be derived from the rotation of the corresponding Wigner distribution function in phase space. In this formalism, the phase factor obtained by McBride and Kerr (1987) is seen to come from the caustics of the harmonic oscillator Green function. Then the idea is generalized to the case of an arbitrary area-preserving linear transformation in phase space, and a concept of the special affine Fourier transformation (SAFT) is introduced. An explicit form of the integral representation of the SAFT is given, and some simple examples are presented.
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Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
The special affine Fourier transformation (SAFT) is a generalization of the fractional Fourier transformation (FRT) and represents the most general lossless inhomogeneous linear mapping, in phase space, as the integral transformation of a wave function. Here we first summarize the most well-known optical operations on light-wave functions (i.e., the FRT, lens transformation, free-space propagation, and magnification), in a unified way, from the viewpoint of the one-parameter Abelian subgroups of the SAFT. Then we present a new operation, which is the Lorentz-type hyperbolic transformation in phase space and exhibits squeezing. We also show that the SAFT including these five operations can be generated from any two independent operations.
We show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.
We give a rigorous proof for the following result that combines those discussed earlier by Kr\"uger and Poffyn [Physica 85A, 84 (1976)] and by Springborg [J. Phys. A 16, 535 (1983)]: The Wigner correspondence of a quantum operator can be obtained by simple substitutions under an arbitrary linear phase-space transformation. This property cannot be generalized to any other transformations or rules of phase-space correspondence. As an application of this result, the Wigner distribution functions of an anisotropic harmonic-oscillator system in crossed magnetic and electric fields are obtained.