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Nonexistence of Finite-dimensional Quantizations
of a Noncompact Symplectic Manifold

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*Mark J. Gotay and Hendrik B. Grundling*

**E-mail:** gotay@math.hawaii.edu, hendrik@maths.unsw.edu.au
**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