Maximal covariance group of Wigner transforms and pseudo-differential operators

Abstract

We show that the linear symplectic and antisymplectic transformations form the maximal covariance group for both the Wigner transform and Weyl operators. The proof is based on a new result from symplectic geometry which characterizes symplectic and antisymplectic matrices and which allows us, in addition, to refine a classical result on the preservation of symplectic capacities of ellipsoids.