A transformation property of the Wigner distribution under Hamiltonian symplectomorphisms


We prove an exact symplectic covariance formula under nonlinear Hamiltonian phase flows (f H t ) for the Wigner distribution \({W\psi}\) . A key role is played in our derivation by the linearized flow at the point where the Wigner distribution is calculated. We show that in general there does not exist any function \({\psi_{t}}\) such that \({W\psi\lbrack f_{t}^{H}(z)]=W\psi_{t}}\).

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Correspondence to Maurice de Gosson.

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de Gosson, M. A transformation property of the Wigner distribution under Hamiltonian symplectomorphisms. J. Pseudo-Differ. Oper. Appl. 2, 91–99 (2011). https://doi.org/10.1007/s11868-011-0023-8

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  • Hamiltonian Function
  • Canonical Transformation
  • Wigner Distribution
  • Weyl Operator
  • Linear Canonical Transformation