Anisotropic uncertainty principles for metaplectic operators
This work establishes anisotropic uncertainty principles for general metaplectic operators acting on $L^2(\mathbb{R}^d)$. It shows that uncertainty is intrinsically directional, confined to an effective phase-space dimension determined by rank(B). Sharp inequalities are derived involving only the directions corresponding to $\ker(B)^ot$, revealing how uncertainty phenomena become directional when symplectic degeneracies are present.