  In this paper we extend the Balian--Low theorem, which is a version of the
uncertainty principle for Gabor (Weyl--Heisenberg) systems, to functions of
several variables. In particular, we first prove the Balian--Low theorem for
arbitrary quadratic forms. Then we generalize further and prove the Balian--Low
theorem for differential operators associated with a symplectic basis for the
symplectic form on ${\mathbb R}^{2d}$.
