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Journal of the Optical Society of America A

Canonical-covariant Wigner function in polar form

Tugrul Hakioglu

Published Dec 1, 2000• 1 citation
Publisher

Abstract

The two-dimensional Wigner function is examined in polar canonical coordinates, and covariance properties under the action of affine canonical transformations are derived.
DOI
10.1364/josaa.17.002411

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We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map \mathcal{S}(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}(\mathbb{R}^{2n}) which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions consists of the translations and the linear symplectic and antisymplectic transformations.
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