JournalsrmiVol. 35, No. 2pp. 317–337

Quantum mappings acting by coordinate transformations on Wigner distributions

  • Nuno Costa Dias

    Escola Superior Náutica Infante D. Henrique, Paço d'Arcos, Portugal and Universidade de Lisboa, Portugal
  • João Nuno Prata

    Escola Superior Náutica Infante D. Henrique, Paço d'Arcos, Portugal and Universidade de Lisboa, Portugal
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Abstract

We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map S(Rn)×S(Rn)S(R2n)\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.

Cite this article

Nuno Costa Dias, João Nuno Prata, Quantum mappings acting by coordinate transformations on Wigner distributions. Rev. Mat. Iberoam. 35 (2019), no. 2, pp. 317–337

DOI 10.4171/RMI/1056