Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold

TL;DR

This paper proves that certain noncompact symplectic manifolds cannot be represented by finite-dimensional matrices, showing fundamental limitations in quantizing these geometric structures.

Contribution

It establishes the nonexistence of finite-dimensional matrix representations for basic observable algebras on noncompact symplectic manifolds.

Findings

No faithful finite-dimensional skew-hermitian matrix representation exists for the basic algebra B.

Finite-dimensional quantizations of Lie subalgebras containing B are impossible.

Results highlight fundamental obstructions in finite-dimensional geometric quantization.

Abstract

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