On the uncertainty principle for metaplectic transformations

This paper deals with a version of the Uncertainty Principle applied to operators in the Metaplectic group, the two-fold cover of the symplectic group. We calculate explicitly the sharp lowerbound occurring in our formulation: we provide a sharp lowerbound for the product of variances of Mu and of u for a function u normalized in $L^2\left(R^n\right)$ and M a metaplectic transformation. The proofs are based upon the symplectic covariance of the Weyl calculus as well as upon some structural facts about the generators of the metaplectic group. We found some motivations in the new proofs and extensions of the Heisenberg Uncertainty Principle introduced by A. Widgerson & Y. Widgerson in [28], developed in [7] by N.C. Dias, F. Luef and J.N. Prata and also in [24], [25] by Y. Tang.