Revista Matemática Iberoamericana


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Published online first: 2019-02-05
DOI: 10.4171/rmi/1056

Quantum mappings acting by coordinate transformations on Wigner distributions

Nuno Costa Dias[1] and João Nuno Prata[2]

(1) Escola Superior Náutica Infante D. Henrique, Paço d'Arcos, Portugal and Universidade de Lisboa, Portugal
(2) Escola Superior Náutica Infante D. Henrique, Paço d'Arcos, Portugal and Universidade de Lisboa, Portugal

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.

Keywords: Wigner distribution, quantum mapping, symplectic covariance, Weyl operator

Dias Nuno Costa, Prata João Nuno: Quantum mappings acting by coordinate transformations on Wigner distributions. Rev. Mat. Iberoam. Electronically published on February 5, 2019. doi: 10.4171/rmi/1056 (to appear in print)