Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold

Mark J. Gotay and Hendrik B. Grundling


Abstract. We prove that there is no faithful representation by skew-hermitian matrices of a ``basic algebra of observables'' ${\fb}$ on a noncompact symplectic manifold $M$. Consequently there exists no finite-dimensional quantization of {\it{any}} Lie subalgebra of the Poisson algebra $C^\infty(M)$ containing ${\fb}$.

AMSclassification. Primary 81S99; Secondary 58F06

Keywords. Quantization, Poisson algebra